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LIFE 



OF 



SIR WILLIAM EOWAN HAMILTON. 



VOLUME III. 



DUBLIN UNIVERSITY PRESS SERIES. 



LIFE 



OF 



SIR WILLIAM ROWAN HAMILTON, 

KNT., LL.D., D.C.L., M.R.I.A., 

ANDREWS PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN, 
AND ROYAL ASTRONOMER OF IRELAND, ETC. ETC. : 

INCLUDING 

SELECTIONS FROM HIS POEMS, CORRESPONDENCE, 
AND MISCELLANEOUS WRITINGS. 



BY 

EGBERT PERCEVAL GRAVES, M.A., 

SUB-DEAN OF THE CHAPEL ROYAL, DUBLIN, 
AND FORMERLY CURATE IN CHARGE OF WINDERMERE. 



VOL. III. 




DUBLIN : HODGES, FIGGIS, & CO., GRAFTON-STREET. 
LONDON: LONGMANS, GREEN, & CO., PATERNOSTER-ROW. 

1889. 



DUBLIN : 

PRINTED AT THE UNIVERSITY PRESS, 
MY PON80NBY AND WELURICK. 




PREFACE. 



nnHE biographical portion of this volume was brought to a 
close and printed early in 1887. This fact will account 
for occasional references in it to persons and events which are 
not in exact accord with the present state of things. The 
subsequent and larger portion of the volume consists of a 
selection from a very extensive correspondence between Sir W. 
E. Hamilton and Professor Augustus De Morgan. To make 
this selection has involved the expenditure of much time and 
labour. The quantity of material was so great that I have had 
to exclude matter that possessed inherent value, either because 
it was in subject unsuited to this work, or because, being mathe- 
matical, the investigations carried on were too abstruse or too 
extended. The general reader will perhaps complain that I 
have introduced more than enough of mathematical investiga- 
tion ; but he will, I hope, withdraw this complaint when he calls 
to mind that it was as scientific men that the writers corre- 
sponded, that it would be unjust to them if their correspondence 
as printed should not retain this character, and that the mathe- 
matical discussions did in fact most often afford suggestion to 
the play of thought which, passing beyond the boundaries 
of science, prompted the wit and the learned and pleasant 
gossip which all readers will enjoy. Its possession of the 
elements last mentioned was the circumstance which determined 
me to place this correspondence in immediate connexion with 
the biography. In some instances I have given a condensed 
statement of the mathematical matter under discussion. For 



VI 



Preface. 



such statements I have, in important cases, been indebted 
to the scientific knowledge of Mr. G. L. Cathcart, M. A., 
F. T. C. D., who has also by his advice and information saved 
me from printing what had either ceased to be of interest to 
the scientific reader, or had been afterwards published by the 
writers themselves. I cannot but take this opportunity of pub- 
licly expressing my gratitude to Mr. Cathcart for a service 
from which the readers of the correspondence will derive much 
benefit, and which cost him a sensible addition to the labours 
of his well-occupied days. 

Another great service to me and to this work demands my 
ample acknowledgment, although that acknowledgment cannot 
now be received by her who had so amply earned it. Miss 
Emily Anne Napier, of whom I feel it an honour to have been 
a friend and distant kinsman, voluntarily undertook to perfect 
for me the chronological arrangement from the beginning of 
the original letters of Sir W. B.. Hamilton and his friends, and 
to make year-lists of them, noting the principal contents of 
each letter. This task, calling for great accuracy, method, and 
perseverance, was fully accomplished by her, notwithstanding 
interruptions from illness and from many other calls of friend- 
ship and benevolence. It was not unlike the service rendered 
by her admirable Mother to her Father, the Historian of the 
Peninsular War, and I am deeply sensible of the extent of the 
obligation under which it has laid me. 

To my friend Dr. Ingram, S.F.T.C.D., I must here express 
renewed thanks for continuing to this volume the invaluable 
advantage which my work has derived from his revision of the 
proof-sheets, and from the indication of errors detected by his 
wide and accurate knowledge. 

In the first volume a note was made of the fact that I was 
indebted to a friend for the Index appended to it. I feel now 
at liberty to state that the friend referred to was Mrs. Edward 
Dowden, and in thus rendering her my thanks for a valu- 
able service spontaneously offered, I am happy to have tln 



Preface. vii 

opportunity of mentioning that in the course of my labours 
I have received unfailing assistance from her distinguished 
husband, whenever I was at a loss as to a poetical allusion, 
or as to any point connected with literature. 

The Index of the present, as of the second volume, has been 
carefully prepared by Mr. T. W. Lyster, B.A., of the National 
Library of Ireland, to whom I am also indebted for most 
efficient aid in drawing up the List of Sir W. E. Hamilton's 
works, and of publications connected with them a list which 
will be found more complete than that of the Eoyal Society. 

I should fail to perform an act of justice were I not, in 
regard to Mr. Gr. Weldrick, of the University Printing Office, 
and his principal assistant Mr. Eooney, to acknowledge grate- 
fully the invariable attention I have received from them, and 
the care they have taken in directing the printing of this work. 
Nor ought I to omit my thanks to the Eeaders and Compositors 
engaged upon it for the pains they have taken to secure its 
freedom from typographical errors. All have worked as if they 
were taking a willing part in honouring the memory of an 
illustrious countryman ; and, although misprints, inevitable in 
any large work, and particularly in one introducing extensively 
mathematical formulse, are to be met with in these volumes, it 
is trusted that they may fairly claim credit for a more than 
average correctness of typography. 

Lastly, I renew the expression of my heartfelt obligation to 
the Board of Trinity College for their liberality in taking upon 
themselves the expense of publishing this work, intended to be 
a full record of the scientific achievements, the varied attain- 
ments, and the personal character, of an eminent alumnus of 
the University of Dublin, and also for their grant made at the 
beginning of my undertaking of the sum of 100 towards the 
payment of amanuenses employed by me during the course oi 
my work. 

Of the imperfections of that work, resulting from my not 
being a mathematician, and from other deficiencies, I am fully 



viii Preface. 

conscious ; but I commit it to the public in the confidence that 
its materials, being the authentic documents of the minds of 
Hamilton and his friends, will constitute it an enduring monu- 
ment of the man as well as of the mathematician, and in 
thankfulness to the Source of all blessings that I have been 
enabled, through a long period not unbroken by illness and 
trial, to carry on to the end my humble part in building it up. 



B. P. G. 



WINTON ROAD, LEESON PARK, DUBLIN, 
March 9, 1889. 



CONTENTS. 



iMlXXOPBINT FBOM A PHOTOGRAPH OF THE BUST BY FOLEY IN 
THE LlBBABY OF TllINITY COLLEGE, DUBLIN, . . . 

PKKFACE, page v 

CORRIGENDA, ,, xxxvi 

CHAPTER XXXIX. 

THE DARGLE LIVERPOOL MEETING VISIT TO WESTMORELAND. 

[185*.] 

PAGE 

Letters from Professor Sedgwick Auguste Comte Theory of Quadrinomes, 
of Tetrads Kirkman Electro-magnetism Dr. Lloyd Engaged on 
numerous trains of investigation Appointment of his nephew to 
Loughcrew Grant from Board of T. C. D. in aid of publication of 
Lectures Dr. Luby and Dr. Lloyd thereon Visit to Dr. Lloyd at 
Kilcroney Dargle Hexameters Speech at Bristol Meeting of the 
British Association: French Mathematics, Abbe Moigno, Professor 
William Thomson, M. Foucault Visit to Windermere : Dovenest, 
Rydal Mount, Mrs. Fletcher of Lancrigg, William Pearson, Mr. and 
Mrs. Richard Napier Carlingford : Mr. and Mrs. Disney Deaf and 
Dumb Boys Lady Campbell: Her books, kindness in life being 
spared, 1 

CHAPTER XL. 

CORRESPONDENCK WITTT AUBREY DE VERE WITH THE NICHOLS CIPHER 

MRS. WILDE. 

[1855.'] 

A. DoVere's Letters on his Conformity to the Church of Rome The 
Nichol family British Association Meeting at Glasgow: On the con- 
c/H.'on of the Anharmonic Quaternion, &c. Visit to Brodick Cnsllr 
His Hamilton descent Sonnet to tny Daughter The Observatory 
visited by Mr. Robert O'Brien and his children * Mary and Alice ! 



Contoih. 



though before that day 'Mrs. Wilde The Problem of Hipparchus 
Patriotic Grief Ellen De Vere and Dora Wordsworth Co UKKSPON- 
DENCE WITH A. DE VERB. Letter from A. De Vere : Old letters, Eliza 
Hamilton, Mrs. Wordsworth, his Sister -From Hamilton : His reli- 
gious views have diverged from A. DeV.'s, but the friendship remains 
From A. De V. : Thanks for this assurance From Hamilton 
on DeV.'s Lines to Burns' J/fi/filand Mary and other poems 
From A. DeV. on his Highland Mary, Farewell to Naples, 
Colonisation, Psyche From Hamilton : His three loves From A. 
De V. : early praise, Confession From Hamilton : Relinquishes 
writing on the past From A. De V. : Anticipations of Future Glory 
and Beauty, The three worlds, Lord Belfast's Poem, Speranza From 
Hamilton : Conclusion CORRESPONDENCE WITH THE NICHOLS. From 
JIdmilton to J. N. : his modest estimate of his own poetry, his Cipher 
From J. P. N. : His estimate of Hamilton From Hamilton to J. 
P. N: Cipher To /. A"..- Herschel, A. DeVere, De Morgan To 
J. P. N. : Parsonstown Sonnet, His daughter, * Despair of the Sea ' 
From J. P. N. ; Friendship in later life, Invitation to Loch Awe 
From Hamilton to J. N. : Indignation at a review of Tennyson's 
Poetry, Scientific Correspondence with Russia, Lord Carlisle's pleasan- 
try, Logic ; necessity of assuming principles, Ptolemy's Almagest, 



CHAPTER XLI. 

VISIT TO CHELTENHAM ICOSIAN CALCULUS CORRESPONDENCE WITH AUBRET 
DEVERE LETTERS ON MATHEMATICAL SUBJECTS. 



Lady Hamilton's illness and his own depression Guest of John Graves at 
Cheltenham, meeting Mr. Grove Meeting of the British Association 
Papers on Scheffler's Der Situations Kalkul and on Gauss's Disquisi- 
tiones Arithmetics THE ICOSIAN CALCULUS The Icosian Game 
CORUE8PONDEKCE WITH A. DE VERE. From Hamilton : Appreciation 
of ssays mostly Theological From A. De V. : Speranza's most 
amiable letter, Sir A. De Vere's poetry, Wordsworth's opinion ot his 
sonnets From Hamilton: Typographical accuracy Estimate of A. 

.'s character From A. De V. : Scotch worldliness not like 
American or English, Defence of the Roman Church, Impressions as 
distinguished from opinions and convictions -From Jlinnf/ftm: 

^antry as to non-version, does not pretend to be unbiassed 
/ //i A. DeV. : Asserted lack of Passion in his own poetry, Ci-itu-ism 
of Henry Taylor, Duty <! rising above impressions From Hamilton .- 

I. K. Robinson Theses on AV>;m/w*w Absorption in matin nia- 
tK.il investigations From A. DeV. : Theological and Matin mat iral 
-i'ii... l'iv ll.in.ili. .M to \vrit l > hik..M.j.hiiMl SuiiiH'K Illustration 



Con fen Is. xi 



PAGE 

from the sea of the Wrath and Love of God, The past a delicate thing 
Hamilton to Dr. Robinson : M. Houel, * Pellipse de M. Hamilton ' 
Hamilton to Dr. Lloyd: Lord Northwick, John Graves's ' Paradise of 
Books,' Copernicus, complex numbers, greatest common measure of 
two quaternions, Gauss's Disquisitiones Arithmetics, extensions to 
Quaternions, Scheffler's Situations Kalkul Letter to J. T. Graves : 
Grassmann's Ausdehnunr/slehre, Moebius, Icosian Calculus, Kirkman 
on Polyedra, 51 



CHAPTER XLII. 

LETTERS ON QUATERNIONS AUBREY DE VERE J S 'MAY CAROLS* BRITISH 
ASSOCIATION IN DUBLIN CORRESPONDENCE WITH SALMON. 

[1857.'] 

Mrs. Smythe of Farmleigh: Linnseus, First chapter of Genesis Letters on 
Quaternions Anticipations as to the Calculus of Quaternions Prints 
for distribution memorandum correcting mistaken ascription to him of 
a mathematical paper by J. T. Graves Vanity CORRESPONDENCE 
WITH A. DE YERE. Letter from A. De V.: J. W. Newman, Robert 
Wilberforce, Venice, May Carols Letter from Hamilton: Essenti- 
ally a Protestant, no scoffer at Popery Letter to Mrs. Wilde : 
Romanism tends to the introduction of a Tetrad into the notion of the 
Divine Nature, this tendency idolatrous, A. DeVere's English Misrule 
and Irish Misdeeds British Association in Dublin under Lloyd's 
Presidency Hamilton contributes a Paper On some Applications of 
Quaternions to Cones of the Third Degree Professor Henry J. Smith's 
concurrent results Communicates also Icosian Calculus and litho- 
graph of Icosian Game Remarks on geometrical and symbolical 
imaginaries Loyalty to his friend Dr. Lloyd Letter from Mrs. 
Lloyd CORRESPONDENCE WITH DR. SALMON. Grassmann, Moebius, 
letters contain some matter worth preserving, his Appendix C., the 
AIM of Quaternions, pleased with Salmon's stout self -confidence 
Letter to Dr. Ingram ; Salmon, Jacobi, Houel, wish as to Salmon, 
Mac Cullagh's quaternion-pleasantry as to Ingram. P. S. of a letter 
to Dr. Salmon. Three things Salmon would admit in regard to 
Quaternions, disclaims the notion of Quaternions superseding deter- 
minants, the two not identical, law for the terms of a descending 
series representing well-known integrals, Quaternions a method of 
Polar Co-ordinates, careless of priority, instance in a formula of 
Henry Smith Lord Brougham, his inaccuracy as to facts, his great 
speech at Grantham on the unveiling of Newton's statue Hamilton 
applied to officially by Lord Monteagle for his opinion on Decimal 
Coinage Drafts a Bill Letter on the subject Paper On the Calcula- 
tion of the Numerical Values of a certain class of Multiple and Dejinitc 
Intct/rah Two long letters on the same subject addressed to De Morgan 
Mode of transition from Double to Quadruple Algebra, . 



xii Contents. 



CHAPTER XLIII. 

FESTIVAL OF POETS VISIT TO EDGEWORTH8TOWN DONATl's COMET 
PROFESSOR TAIT BURNS* CENTENARY. 

[1858.] 

PAGE 

Growing size of the Elements of Quaternions Festival of Poets Torquay 
Describes singing of birds Aubrey De Vere's religious earnestness 
H. Taylor's poetical tribute to him Professor Hennessy, Dr. Waller, 
Dr. Anster, Mr. Grenell The Earl of Carlisle Visit to Edgeworths- 
town Sonnet, 'To Mary Edgeworth' Dangerous illness of his 
daughter Hon. Mrs. Ward Professor Tait introduced to him by 
Dr. Andrews of Belfast Letter from Tait on Quaternions From 
Dr. Robinson Hamilton's opinion of Tait's work in Quaternions 
Descartes Invited to attend the Burns' Centenary Festival at Glas- 
gowLetter on Burns, 97 



CHAPTER XLIV. 

CALVINISM DATING OF SCIENTIFIC WORK PORTRAIT HERSCHEL ON QUATER- 
NIONS M ANSEL'S BAMPTON LECTURES BRITISH ASSOCIATION AT ABERDEEN. 

[1859.'] 

Hamilton's opinion of Calvinism Letter to Professor Jellett on accuracy 
in recording dates Mansel's Bampton Lectures Letter to Dr. Lee 
Science of the Infinite Visits Rhyl Visits with his daughter the 
Moravian establishment at Fulneck Attends Meeting of the British 
Association at Aberdeen Meets Clerk Maxwell On an Application 
of Quaternions to the Geometry ofFresneVs Wave- Surf ace Quater- 
nions applied to Physical Hypotheses Visits Balmoral Death of Dr. 
J. P. Nichol Lectures at Fulneck Portraits of Hamilton Sir J. 
Herschel's Cry of Distress Herschel's satisfaction with Hamilton's 
reply, Ill 

( IIAI'TKI; \\.\. 

INHARMONIC COORDINATES GEOMETRICAL NETS IN SPACE GEOMETERS IN 
i IV COLLEGE CIRCLE-SQUARING AUBREY DK VERB'S < INISFAIL.' 

[18*0.} 

Correspondence with l>i. II nt on An harmonic Coordinates Geometers in 
Trinity College Mr. James Smith the Cirole*8quarer 



Content^. xiii 



PAGE 

Pi-onfthnt Kiyht 7Y/-////r/V>-.s nf tin- Insrrfb,-(/ Pali/yon ofTwenti/ .S'/VA-.s 
exceed Twenty-Jive Diameters Death of Archiaima Hamilton Visits 
the county of Wicklow COKKKSI-OXDENCE WITH A. DEVEKE. Letter 

from A. DeV.: D. F. Mac Carthy's translation from Calderon's 
'Autos Sacraraentales ' Letter from Hamilton: Dr Yen-'* 'Select 
Specimens of the English Poets,' Sir A. De Vere's Tragedies Letter 

from A. DeV.: His own poem * Inisfail,' Irish History, . . . 1'2:J 



CHAPTER XL VI. 

RECOGNITIONS FROM ABROAD HONORARY DEGREE AT CAMBRIDGE TAIT 
INGLEBY KIRKMAN. 

[2M1.] 

Bellavitis Meibauer Accepts from Dean Graves a Vice- Presidentship 
of the Royal Irish Academy Oifered by the Vice-Chancellor of the 
University of Cambridge the Honorary Degree of D. C. L. Hesitation 
in acceptance Receives the honour in distinguished company 
Presented to the Prince of Wales Visit at the Observatory from 
Professor Tait Miscalculation as to the time which his work on the 
Elements of Quaternions would occupy, and the magnitude of the 
book Relations of Hamilton and Tait as authors on the subject 
Comet Social Science Congress Lord Brougham Makes the ac- 
quaintance of Dr. Ingleby Attends Meeting of the British Associa- 
tion at Manchester Visits Rev. T. P. Kirkman His estimate of 
Kirkman Devotion to study Criticism of a passage in D. F. 
MacCarthy's translation of Calderon's ' Love the Greatest Enchant- 
ment,' .129 



CHAPTER XLVII. 

EXPENSE OF PRINTING ROBER'S ESSAY AMOUNT OF SCRIBE-TOIL. 
[1862. ~\ 

Alarmed by the cost of printing the Elements Consults Dr. Hart His 

idlings towards Dr. Salmon The Board advances a second 100 

T'S construction of the Heptagon Dr. Haughtou and Conical 

Refraction Homer's Method Hamilton now a solitary worker His 

Aunt Sydney His Sons, 138 



xiv Contents. 



CHAPTER XLVIII. 

SCIENTIFIC ITEMS CONCLUSION OF THE DEVERE CORRESPONDENCE THE 
ROYAL IRISH ACADEMY AND THE ROYAL DUBLIN SOCIETY. 

[1863.] 

PA 

Letter from Tail : Conical Refraction, Prof. Bolzani, enormous value of 
Quaternions to Physicists Letter from Hamilton : Energy and Work, 
Negative foot-pounds of work done by gravity Frost and Wolsten- 
holme's ' Treatise on Solid Geometry ' Osculating twisted cubics 
General centre of applied forces His originality attested by Dr. 
Salmon Advice respecting Scientific work to his son Archibald 
Beyers' contracted multiplication Herschel's anticipation of Hamil- 
ton's triumph CONCLUSION OF CORRESPONDENCE WITH AUBRKY 
DE VERB. Letter from Hamilton : Experience of mathematical work 
during pain. Letter from A. De V. : The heart never grows old 
Hamilton' ' last Utter to A. De V.: De V.'s ' Hymn for Good Friday,' 
sympathy with him in the most vital doctrines of Christianity. 
A. De V. J s last letter to Hamilton Sonnet by A. De V., In memory of 
Sir William Rowan Hamilton, after reading again his letters 
Memoranda by Hamilton respecting the Meeting of the Royal Irish 
Academy, July 6, 1863, 149 



CHAPTER XL1X. 

EXCURSION IN WICKLOW DR. INGLEBY AND METAPHYSICS HAMILTON'S 
POETRY DECLARATION OF MEN OF SCIENCE. 



Extreme diligence in mathematical investigation and correspondence 
Excursion with his daughter to the county of Wicklow Reminiscence 
of visit with Wordsworth to Glendalough in 1829 Dr. Hart consults 
Hamilton as to Plana's investigations regarding solar heat at different 
points of the earth's surface, and his own results on the same subject 
Hamilton bears testimony to Dr. Hart's talent for simplification- 
Statics really a part of Dynamics Foucault's Pendulum Experiment 
The Plumb Line Experiment Dr. Hart suggests leaving physics 
for a second volume of the Elements Retracts the suggestion 
Hamilton again anxious about cost of publishing Letter to Dr. 
Robinson on the subject Robinson's reply Calculation as to date of 
the Hejira Wordsworth's claim to be descended from King Alfivd 
Turns to Metaphysics C<M:I;I M'<>M>I;NCI: WITH Du. IMM.KHY. Ego 
and Non Ego, Perception, Walter Scott's mistake about the Septuagint, 
Sense of Sight, Sense of Touch, Coleridge on Berkeley, Sir W. JIamil- 



Content* xv 



ton of Edinburgh, Hamilton asserts that he sees distance, Dr. Ingleby 
on Reciprocal Causation, Descartes Letter to Mr. Barlow. Hamilton's 
Reid, Hallam, De Morgan, Des Cartes' * Eao cogito, ergo sum' 'My 
only power is mathematics ' Note : Hamilton on Wordsworth and 
Tennyson Hamilton's Poetry Mourey's relation to Quaternions 
( 'i pi ici- Declaration of men of science Clerk Maxwell Hamilton's 
value for intellectual freedom, 163 



CHAPTER L. 

CAYLEY HAMILTON'S LATEST SCIENTIFIC WOEK LAST ILLNESS 
HONOUR FROM AMERICA DEATH. 

[1865.'] 

]"< ling with which he enters on his last year Sonnet by his younger son 
Intentions as to conclusion of the Elements of Quaternions Profes- 
sor Cayley : ' Principle of Equivalent Moments ' Dr. Salmon's con- 
fidence in Hamilton's mathematical accuracy Hamilton refers to 
Professor Jellett for testimony as to Mac Cullagh's having arrived at 
the Principle in question Jellett testifies that he had Cayley accepts 
the rectification of his enunciation Modification of last sentence 
printed in the Elements of Quaternions Dean Graves's Theorem on 
Binomial Coefficients Hamilton's generalization of it His last con- 
tribution to the Proceedings of the Royal Irish Academy : * On a New 
System of Two General Equations of Curvature . . . deduced from 
Gauss's 2nd Method, &c.,' and * Three Verifications of Measure of 
Curvature ' Note : Record of investigation in manuscript Illness 
The printer's bill Visits the International Exhibition in Dublin 
Attack of gout Rallies and resumes work Kind neighbours Honour 
from America Placed first on the list of Foreign Associates of the 
American Academy of Arts and Sciences Official Letter from B. A. 
Gould, private letter Hamilton's letter to Dr. Hart communicating 
the intelligence Dr. Salmon's congratulatory letter Hamilton's 
acknowledgment to B. A. Gould Last illness Death Funeral 
Inscription on Headstone Surviving members of his family, . .193 

CHAPTER LI. 

CHARACTERISTICS OF HAMILTON AND OF HIS WORK. 

Letters of condolence from Sir J. F. W. Herschel and Professor Augustus j 
De Morgan Extracts from brief Memoir by De Morgan Hamilton 
the ' Irish Lagrange.' Fourier's characterisation of Lagrange Dean 
Graves's Eloge Corroboration of it by Herschel Eloge by Rev. 
Charles Pritchard, President of the Royal Astronomical Society 
Article by Professor Tait in North British Review Memorial Sonnet 
by Aubrey De Vere, ( Friend of past years, the holy and the blest,' . 214 



xvi Contents. 



CHAPTER LII. 

(SUPPLEMENTARY.) 

A GATHERING OF FRAGMENTS. 

Applies to himself Henry Taylor's character, in Edwin the Fair, of 
Wulfstan Hamilton not unpractical Applies to himself Ptolemy's 
characterisation of Hipparchus and Juvenal's precept for a wise man's 
prayer His claim in regard to causing pain to others True ground 
for tolerance Sonnet at the close of a period of solitude, ' Draws to 
an end a melancholy while ' His wife's weak health ' An abstract 
idea ' His domestic affection His interest in children Instance 
His truthfulness Instance Kindness to dumb animals Incident of 
the Dove Physical courage Instance Once contemplated a duel 
Alleged interchange of compliments between Wordsworth and 
Hamilton Ruled paper Mathematics applied to Metaphysical 
reasoning, Coleridge, Kant Quaternion stanza in Beattie's Minstrel 
W. E. HAMILTON'S MEMORANDA OF HIS FATHER'S HABITS : Prepara- 
tion for Lecturing, Procrastination, Illustration, Inspiration of Scrip- 
ture, Resurrection of the body, Duty of laymen as to advocacy of 
religion Professor Sylvester * On Hamilton's Numbers,' . . .228 



CORRESPONDENCE BETWEEN SIR W. R. HAMILTON AND 
PROFESSOR AUGUSTUS DE MORGAN. 

I -ATKMKNT, 



From Date of Letter. 

1841. 

A. l'< M. May 8. Presents his Paper On t/t<- Foundation ./' Alijt'lra us 

introduction. Asks for Triplets. 

\\ '. ll.ll. May 12. Algebra the Sou -m-i ,.l I 'ure Time. Kant. P.S. Triplets, 
1842. I'olypleta. 

A. l>. M. Jan. '{. Hun-Mil. Logometers. 1 >i\truvnt Series. 

\V. 11. 11. Jan. us. Second Essay on Dynamics. 

A. J).-.M. .l.in. :J1. lnvitrs Ili-puri on Hanson. 

s. Inv. Mirations on a dillriviit linr from llaii.^ii^. Fluc- 
1844 tuating Functions. 

<1 i lap. r .MI Trij.l, Aluvl.ru. 



Contents. xvii 



From Date of Letter. 

W. li. II. Dec. 9. Does not refuse to triplicise. Challenges any other theory 
of Pure Triplets to surpass in thive points that in- 
cluded in his Quaternions. BymmetrioalneM of > 
the most important. 1st P.S. Solution of prolilmi 
as to resultant of forces, &c. 2nd P. S. Argument 
against Triplets reconsidered. 3rd P.S. Would 
not prevent the study of Pure Triplets. 

A. DeM. Dec. 16. You all for interpretation. My object to keep to symbo- 
lic rules. Suspects you have the right sow \\\ UK- 
ear. 
,, Dec. 30. Fractional Differential Coefficients. Cautions as to health 

1845. and work. 

W. R. H. Jan. 5. Two notes. Geometrical interpretation of A. DeM.'s 

Cubic System. 
A. DeM. Jan. 12. Acknowledgment. 

,, Dec. 23. Asks for information as to H.'s quarto publications. 

Advises to give correct information to bibliographers. 

1846. Mr. (C.) Graves's Paper. H.'s Symbolic Geometry. 
W. 11. H. Feb. 2. Quaternions u a new algebraical geometry." Intention 

to retire from Presidentship of R. I. A. Time of 
Easter. Equinox in the year of the Council of 
Nicsea. 

A. DeM. Feb. 15. C. Graves's Paper a finished thing. Quaternions a system 
in which rotations play the part of co-ordinates. 
Approves of resignation of Presidentship. Easter. 
Burckhardt. Chernac's Cr thrum Arithmetical m. 
Mistake of Gregorian Committee. Arbogast's 

1847. method of derivations. 

,, April 12. Appeals from a charge of piracy in Logic brought against 

him by Sir W. Hamilton of Edinburgh. The two 
Sir W. Hamiltons constantly confounded. 

W. R. H. May 7. Researches respecting Quaternions. Manner in which 
the mathematical notion of Time leads to a general 
conception of Numerical Sets. Mind on the present 
subject less analytical than synthetical. " Geome- 
trical Fraction " of one set of Papers the same as 
" Quaternion " of another. Offers Paper on Law of 
Circular Hodograph. Interested at present in some 
applications of Quaternions to Physical Astronomy. 

,, July 10. Death of his Uncle James. Coincidence. Hopes that 

Quaternions will simplify physical astronomy. 

A. DeM. July 12. Is there a mental dynamics ? Action of Moonlight. 
Stories of ghosts. No doubt of the applicability <>r 

1848. quaternions. 

,, Aug. 17. Appeal for Second Part of Memoir on Dynamics. 

W. K.il. Aug. 21. Feels that he ought to write a quaternionic treatise on 
mechanics. Pressed to write a bock on Quaternions. 
VOL. in. b 

Jn, 



XVI 11 



Contents. 



Solution of x*-1x= 5 



W.R.H 



Aug. 
Sept. 24. 



A. DeM. Sept. 26. 



From Date of Letter. 

1849. 
A. DeM. April 28. Mr. Wentworth Erck. Co-cosine. 

to 103 decimal places. 
30. Professor Young's loss of his post at Belfast. 

Professor Young. Commissioner Hargreave. Peacock's 
observation that Functions and Quaternions have 
several analogies. Infinitely many solutions of 

fm-l. 

Professor Young. Hargreave. Algebra itself a func- 
tional calculus. Some loose thoughts (algebraical). 
Parentheses. Chancery suit. Swing-swang integra- 
tion. 

Answer to one of the " loose thoughts." Mr. Hargreave. 

" Closed circuit " question. Illustration of boys and 
beadle. 

Some electrical phenomena of circuits not unlikely to 
correspond with A. DeM.'s law. Theorem respect- 
ing Gauche Polygons. Curious to see a demonstra- 
tion of either theorem by co-ordinates. 

A. DeM. Oct. 11. A searcher for things mental not material. Pascal's 
theorem about the hexagon. 

An enigma for W. Si. H.'s daughter. 



W. R. H. 
A. DeM. 



Oct. 
Oct. 



W.R.H. Oct. 10. 



W.R.H. 
A. D.M. 

W. R. H. 



Nov. 12. 

1850. 

April 6. 

April 9. 
April 14. 

April 18. 
Oct. 19. 

1851. 
Nov. 26, 



A. DeM. Nov. 26. 



W.Il. II. 
A. DeM. 



I I.e. 

DM, 



Relatives. Motto of Cambridge and Dublin Mathemati- 
cal Journal. 

Generalisation of A. De M.'s theorem. 

[Two Notes.] The two Sir. W. Hamiltons reciprocal 
polars. More on the theorem. 

Continued discussion of the same theorem. 

The Calculus of Quaternions includes all ordinary 
Algebra. Points in which Algebra and Quaternions 
are distinguished. 

Thanks for Paper on "Recent Discoveries on the Inven- 
tion of Fluxions." Vindication of Leibnitz from 
charge of plagiarism gives him pleasure. Suspects 
he has been plagiarised from. Book on Quaternions 
advancing rapidly. Asks for references to works 
having connexion with the subject of quaternions. 
Relations between Double Algebra and Quaternions. 

Reason for shrinking from the subject of Triplets. 
Death of F. Baily. Sir W. Hamilton of Edinburgh. 
Quaternions must contain the double algebra. 
Plagiarism. Omission in published list of W.R. 1 1 . '* 
Time paper. Wheatstone an authority on Germi 
writers. Leibnitz. Laplace. 

Use of It - A in geometry. Win at *tone. 



Contents. xix 



From Date of Letter. 

W. It. H. Dec. 8. Asks for criticism of the metaphysics of the early articles 

of liis Lectures. 
Dec. 10. Object of Lectures. A fatal fault in Double Algebra. 

A. DeM. Dec. 15. Acknowledgment. 

W. R. H. Dec. 16. Despairs of writing anything good on the history con- 
nected with quaternions. 

A. DeM. Dec. 17. Will set you at rest about the history. 

\V. K. H. Dec. 18. Fears a mistake in a quotation from Persius. Walker's 
Logic. 

A. DeM. Dec. 20. Quotation correct. Scientific poems. 

\\.R.H. Dec. 23. Walker's Logic. John Walker. 

,, Dec. 24. Has in the last year composed only four sonnets. Could 

never write verses on the death of a blood-relation. 

,, Dec. 24. Depot. French accents. Greek accents. English spel- 
ling. 

A. DeM. Dec. 27. Thanks for Sonnets on Tetractys and Garden. John 
Walker. Depot. Accents. Peacock's Report on 
Analysis. Gauss. About 1831-33 there was a 
revival of pure algebraical speculation. The gener- 
alities of Sir W. R. H.'s paper incapable of being 
transferred elsewhere. Special features of quater- 
1852. nions. 

W. R. H. Jan. 3. Polyplets. Geometry at the last moulded and fixed my 
conception of ijk, yet abstract speculation led to it. 

A. DeM. Jan. 4. Find Quaternions no child's play. All that is peculiar to 
Quaternions beyond the reach of anyone except 
yourself to claim. Nothing in Gauss to suggest 
quaternions. Ohm on Algebra, translated by Ellis. 
Give definitions. Your alleged defect in double 
algebra a defect in symbols. 

W. R. H. Jan. 5. Encloses order for Ohm's book. 

,, Jan. 6. Private reason for knowing that Gauss had not antici- 

pated quaternions. Baron von Walterhauseu. 
,, Jan. 7. Acknowledgment. Up all night for eclipse. 

A. DeM. Jan. 7. Mathematicians and amusements. Mathematical book- 
sellers. Gauss's Disquisitiones Arithmetics. J. 
Walker. Gauss's Quadratic residues. 

W. R. H. Jan. 8. Style of Lectures on Quaternions. Do not assume much 

previous knowledge of mathematics. 

,, Jan. 10. Gauss's quadratic residues. Solution by quaternions of 

the topical problem of Apollonius. 

A. DeM. Jan. 10. Lends Mourey. French likely to claim quaternions for 
Mourey. 

W. R. H. Jan. 13. Servois nearer than Mourey to such an anticipation. 
Argand. Fransais. Buee. P. S. Mourey's proof 
of the existence of a root. History of \/ 1. 

b2 






Contents. 



Date of Letter. 

A. DeM. Jon. 15. Criticism of Servois. Mourey nothing else but a part of 

double algebra. 

Jan. 18. Criticism of Sonnets. Functions of in common algebra. 
Instance. History of \/- 1. 

\V.K. H. Jan. 21. Combination of science and imagination personified in 
Herschel. Careless of criticism of my poems. Sensi- 
tive as to my mathematics. 

A. DeM. Jon. 22. Geodetic lines observed in Dublin Observatory. Asks 
for description for Annual report of Astronomical 
Society. Priority question fully discussed. 

\\ . K. H. Jan. 26. Hopes as to Quaternions, and opinion of their present 

state. Gauss's biquadratic residues. Kummer. 
Jail. 26. Origin of the word " Panic " from Polysenus. 
,, Jan. 27. Change of mode of address. Lady Campbell. Euclid in 
Greek. Bishop Butler's observation about caution 
to avoid being mistaken. 

A. DeM. Jan. 28. Pan's horns. 

W. R. H. Jan. 31. Suppresses suspicion as to plagiarism on himself. Gener- 
ous treatment by eminent men. Explanation as to 
himself and the Graveses in relation to De Morgan 
and Triplets. Mac Cullagh and Quaternions. 
Polysenus's Stratagems. Once invented a Lucianic 
dialogue about the Rape of Helen. " Reviewers 
Reviewed." 

A . I'M. Feb. 2. Suppose your race susceptible as to priority. His own 
nativity. Mulcahy on the altitudes of modern 
geometry. Murphy. Salmon. Geometry taken 
root in Ireland. Suspicion of others copying infe- 
rentially. 

\V. 1; II. Feb. 9. Proof of a theorem of Desargues. Pure geometry in 
Ireland. Mulcahy. Eton and Learning. Irish 
susceptibility. Prejudices against Quaternions. 
Anecdote. MacCullagh's question. Salmon. 

A 1'. M. Feb. 12. The equilateral hyperbola. The ellipse. Modern geo- 
metry worked on a system of signs. Salmon' 
working of general propositions. llulieulers of 
Quaternions. MacCullagh's question. Objects to 
phrase " Six points in involution." 

;. De Morgan " On the Connexion of In volute and E volute in 
Spare." Bull at the Castle. Allinan, .1 ukes, Anster, 
I ly B mi-. 

! I. Kurli.l. Tliuli's. The i.jnilat. ral hypcrltoloid. Coiu- 
n Dr Merlin's |' ;i p rr "On (he Connexion, 

&C." Tlir rirrlr Hi l a (|(-clUlatr rllipM', lillt 



Contents. xxi 



From Date of Letter. 

A. DeM. Feb. If*. The Circle and the Ellipse. Self-similarity in both. 
Also in the Screw, which it was a failure of Euclid 
not to introduce. Cylinder. King. Self-cut 
Ring. Incloses note to Mr. Salmon really a valu- 
able writer. Lady Kane. 

W. K. II. Feb. 21. Mr. Salmon. [Letter from Mr. Salmon of Feb. '21: 
Proof of anharmonic theorem, readiness to employ 
the most convenient instrumentality. Quaternions 
now a bow of Ulysses which no one can bend but the 
owner. Reply by W. 11. H. : Proposes comparative 
exhibition of different mathematical methods.] 

A. Ik'M. Feb. 23. A table of contents a palisade against mere talkers. Irish 
more given to pure speculation than the English. 

\V. K. H. Feb. 25. Difficulty in printing. 

,, Mar. 15. Is the word " eliminant " your invention ? 

,, April 2. Speech in honour of Moore. His "rebellious" poetry. 

No harm in it. 

,, April 15. Mesmerism. Miss Edgeworth, Herschel. Believing one's 
eyes. Conical Refraction. Dumas' "Memoiresd'un 
Medecin." Lord Clarendon, Babbage. 

A. DeM. April 15. Grant's History of Physical Science. Differential Calcu- 
lus. Laplace. Fermat. 

W. R. H. April 16. Southey's saying about proof-sheets. Sydney Hamilton 
of Trim, infant pupil in Sanscrit and Persian. 
Mathematical examination papers. 

A. DeM. April 16. Clairvoyance. Good notion of fluxions from Oresmius, 
Episcop. Lexov. 

W.R.H.*Aprill9,20. Walter Scott on extraordinary events. Herschel's 
* Collection of Examples.' A theorem. Astrology. 
Algebra of bi-couples. 

,, * April 22. Definition of the differential of a function of a quaternion. 
Equation in quaternions representing an arbitrary 
surface of revolution. 

A. DeM. *April23. Comment on definition of differential. Suggests another 
mode of differentiating a function. Will require a 
year after book is out to follow you. Sir W. 
Hamilton of Edinburgh. ' Rowan.' Astrology. 
Guy ManncrliKj. Alexandrian Sorcerer. 

W. K.II. *April26. Comment on A. DeM.'s method of differentiating a 
function. Challenge. Three advantages of his own 
method. 

A. DeM. * April 27. Sceptical about ( + i), &c. A difficulty. When both wand 
ti are fractional what possible form of development ? 

W. R. H. *April 29. No possible form of development analogous to the 
one. In quaternions there is a development. 



* Letters with an A>U'risk are to he found at 



xx ii Contents. 



From DateofLetUr. 

A. DeM. April 30. Doubts about <". There is a development, but which of 

an infinite number ? 
May 1. Reply to yours of April 19. Cannot interpret W. Scott 

on extraordinary events. Instances of coincidences. 

Astrology, w 3 - n intruding roots. Cardan's result, 

page 623. 
May 3. Geometry and its algebra in T. C. D. Punishment for it 

in the next world. 
W. U. H. May 3. Determination of the differential of the square-root of a 

quaternion. 
,, May 5. Priority. Proposes to cite DeM. Not pleased with 

Bronwin's citation of * Hamilton's Theorem.' ' Curse 

of Kehama.' Latin poem by Leibnitz. 
A. DeM. May 6. Walker could think about logic. My own claim. Partial 

interpretations may solve problems. ' Hamilton's 

Theorem,' Herschel's. The extension Hamilton's. 

Citation. Letter to Sir W. H. of Edinburgh. 

v7roffTa(ris, ovvia, xopo/CTTjp. Heb. I. 3. 

W. R. H. May 8. Heb. I. 3. Walker's Logic. 

May 9. Henry Crabb Robinson. Herschel's Theorem ? 

A. DeM. May 10. Herschel's Theorem. Heb. I. 3. Intends to apply the 
theory of probabilities to St. Paul's alleged authorship 
of this Epistle. 

AV. R. H. May 12. To whom are divine honours to be paid ? Regrets having 
entered into religious controversy. 

A. DeM. May 16. De Morgan not de Morgan. Reason for not taking M.A. 
degree. Worship of Jesus Christ, meaning of ' wor- 
ship.' * Honour.' Anecdote. Porismatic system. 
Libri. 

,, May 21. \/~~l * multiple interpretation. Whewell's name for 
interrupted inversion. Wanted a short and sound 
interpretation of a theorem. Fermat's Method of 

maxima and minima. Kepler. 



W. K II. May 24. Multiplicity of interpretation of ( -J " DeM.'s proof of 

a cited equation satisfactory. 

May 26. Possibility of a visit to Greenwich. In 1838 climbed into 
the Ball at St. Paul's. Effect of the wind. Coinci- 
dence with an expression of C. Dickens. Another 
coincidence with an expression of Tennyson. Dinner- 
rhymes. Biquaternions. 

A. DeM. May 27. AVill not be at Greenwich. Dinner-rhymes. Cannot 
keep to Swift's vein. First surgical operation under 
mesmerism. 

I .tli uu Asterisk are to be found at page> '-.; 632, 



Content*. 



xxin 



A. DeM. June 8, 9. <p.r = 



W.H. H. June 12. 



A. DeM. 



June 15. 
June 20. 



W. R. H. June 23. 



From Date of Letter. 

A. Dr M. May 31. Shadow of a pole at sunset. 

W. R. H. June 2. A new sort of variations of definite integrals. Cauchy. 
Libri. Fellowship Examinations conducted in 
English. 

Partial solutions of equations. Algebra not 
made for geometry. Double algebra is of a surface, 
single algebra linear, double areal. Quaternions 
with the three rotations ought to be the triple 
algebra. 

Results respecting a particular sort of variation of a 
definite integral. Investigations respecting differen- 
tials of functions of quaternions more general than 
that respecting the differential of a square-root. 

Men to whom Lectures on Quaternions should be sent. 

Dr. Logan. Libri. Kirkman's mathematical mnemonics. 
Catechism of mathematics. Libraries. Airy, prince 
of methodists. 

Expectations in connexion with Quaternions as regards 
electricity, magnetism. Polarity. Mnemonics. A 
formula hard to be remembered. Politics. 

Lodges a paper on Syllogism considered as a composition 
of relations. A play on words. 

Oppressive heat. Chasles : Geometric Superieure. Liou- 
ville's Monge. Poinsot's Theory of Rotation. Justin 
Brenan v. Logic. 

Dublin election. Beautiful aspect of city and river. 
The Bi-tensor of a product of Biquaternions. 

Politics. My reform of the House of Commons. 
Theory of decision by votes. House of Lords. In 
English hands, not objecting to a moderate mixture 
of Irish and Scotch, anything would work well. 
Macaulay on the House of Commons. Libri. Prince 
Boncompagni. Gerard of Cremona, Translator of 
Ptolemy. Negative quantity in Arabic algebra. 

oo = 10 13 . Reinhold's Tabulce Directionum. My Mnemonic 
for the hard-to-be-remembered formula. Suggests a 
sonnet. Libri. 

Politics. Robert Button. T. Drummond. The Ballot. 

The Times on Libri. Jewels of criminal law in France. 
Prof. Young. Maynooth. 

Dublin politics. 

Admits deep prejudice in politics. Reminiscences. Claim 

of family descent. Maynooth. 

A. DeM. July 27. Politics. How I would tinker the Constitution. Evil of 
subscriptions. Opposition to Maynooth grant founded 
on a subversive principle. Reinhold's Tabula 
foscunda. 



A. DeM. 



W.R.H. 



A. DeM. 



July 1. 
July 11. 

July 12. 
July 14. 



July 1C, 17. 



W.R.H. 
A. DeM. 

W. R. H. 



July 20. 
July 23. 

July 23. 
July 26. 



xxiv Contents. 



From Date of Letter. 

W. U. II. .1 uly US. Against multiplying oaths. Bribery. 

A. DeM. July 31. Epigram on the Duke of Norfolk and curry powder. 

Your Seal. Lucas Pacioli's Sumtna di Arithmctica. 

The last day of letters prepaid in money. 
W. II. II. Aug. 3,4. Calculation unfavourable to the notion that the asteroids 

are fragments of an old planet. Such calculations 

not his most useful employment. His birthday and 

his younger son's. 
Aug. 8. Lithograph of C. Wolfe's autograph of ' Not a drum was 

heard.' Preface. First or Third Person ? Lady H. 

and his daughter desire to see De Morgan. 
A. lt M. Aug. 9. Asteroids probably parts of a ring. Bibliography. 

Errors in Description. 
,, Aug. 10. Keasons for First Person. Advice as to Preface. Scotch 

Ancestry. Dr. Parr to Sir James Mackintosh. 
W.ll. H. Aug. 11. Two Solutions of your puzzle for my daughter. Her 

birthday. 

,, Aug. 13. Chasles on imaginaries. Time of publication of book. 
A. DeM. Aug. 16. Only two solutions of the puzzle. Dates. 
W. R. H. Aug. 20. Paper on the Ascension. As Trinitarian as Athanasius. 

Sermon by Salmon. 

A. DeM. Aug. 23. F. Baily's correspondence. Your handwriting diminish- 
ing. Your Trinitarianism. Your ' sermon.' Mr. 

Robinson and belief in the Devil. Mrs. Flamsteed's 

Valentine. 

AY. R. H. Aug. 26. Ely Sonnet. Sir J. Herschel's to Hamilton. 
A. DeM. Aug. 30. The Wolfe lithograph. A Peacock a Phoanix. The 

Three Churches in the Ely Sonnet. Herschel a man 

of poetical elements. 
W.R. H. Sep. 1. Explanation of Ely Sonnet. Bishop Terrot. Abp. 

Whately and the memorial window in Cast K knock 

church. 
A. 1>< M. ,S-p. 1. Had practice of printers putting books a year forward. 

Theology. An agreeable lull. 
\V. It. II. Sep. 2. Satisfaction about the Wolfe autograph. My Preface a 

bore. 
A. ! M. N p. 3. Paradise Lost necessarily Tritheistic or Arian. Whnti-ly 

and emblems. I would not abolish SHJH -rstition ami 

mysticism. My notions about burial. 
\\ l;. II. .-. p. 10, 17. Belfast Meeting of British Association. Carlin-tonl. 

l;< newed interest in Quaternions. 

A. DeM. Sep. 20. Given X-land, what constitutes an X-man ? Chamvllor- 

of Oxford. Walton, opponent of I'M ik< 

son of Isuak W. P 
l;. turn to Dublin, 

A. I- p. 27. iwtfolios. History of infinitesimals in England. 

V wton. 



Contents. xxv 



From Date of Letter. 

\V. It. II. S(j>. 2'J. Salmon on Temporal Blessings. National Sins and .Judg- 
ments. Irish murders. Gift of a Douay Bible. 
Charitable supposition of II. C. Priest. 

A. Di-M. Sep. 29. Enquires for Boole about Dublin printers. Check upon 

printers. 

,, Oct. 1. Salmon's argument that religion was intended to develop 
the inquisitive faculties sound, but involves the con- 
sequence that differences of opinion were also 
intended. A difference with Salmon. Irish Exodus 
pun. Answer to application for subscription 
towards a R. C. purpose. 

W. K. H. Oct. 3. Dublin University Press. Boole. 

,, Oct. 4. Bishop Berkeley. Walton. * Fluctions ' and ' fluxions.' 

A. DcM. Oct. 5. Irish bulls. Boole has got hold of the true connexion of 
algebra and logic. 

\\Ml.H. Oct. 8. Gauche Polygons. 

A. DeM. Oct. 23. Editions of Berkeley. Infinitesimals in England. Newton. 
Four colours will colour any possible map. 

\V. K. H. Oct. 26. Asks for a sketch of the history of Double Algebra, 
-v/- 1. Grassmann. 

A. DeM. Oct. 29. Cannot comply with request. Grassmann (Nebuchad- 
nezzar). Sir W. H. of Edinburgh and his charge 
about Ploucquet. Translate for me your German. 

W. K. H. Oct. 30. Begun Preface-writing. 

A. DeM. Nov. 8. Series and mean values. 

W. R. H. Nov. 16, 18. Triplets worked at in 1835. Right now to record this in 
Preface. 

A. DeM. Nov. 20. Admits his right. Not at the Duke's funeral. The 
procession. 

W. R. H. Nov. 24, 25. Your Triple Algebra and my Triplets. Fundamental 
difference between them. My notion of triplets as 
old as 1830. 

A. DeM. Dec. 15. Do not be uneasy about priority. Grass and mutton. 

\\ . K. II. Dec. 18. Admits nervousness as to anticipated controversy. When 
it comes the vexation is over. Trisection of an angle. 
Is it impossible by Euclid ? Gauss. 

A. DeM. Dec. 24. Your priority-vexations a reflection on Lady Hamilton. 

Proposed resolutions of British Association. Gauss's 

discovery of a method of inscribing a pobrgon of 

seventeen sides increases my disbelief in possibility 

1853. of trisecting an angle. Tells a story. 

\V. 11. II. Jan. 8. Argand true author of Double Algebra, not Buee. Asks 
for any information to the contrary relative to BmV. 
,, Jan. 20. Double Algebra. Bute, De Morgan, Wallis. 

A. DeM. .Ian. -7. lUu'e, \Vallis, \\arreii, Am ai'raid "i virtual discoveries 



XXVI 



Con It ii fs. 



Feb. I. 



A. DeM. Feb. 5. 



W.R.H. Feb. 9. 



A. DeM. 



W. R. H. 



Feb. 11. 
Feb. 12. 



April 
April 



Date of Letter. 

\\ . K. H. Jan. 28, 31. J. T. Graves confirms recollections about Argand. Buee, 
Warren, Wallis. Been reading Grassmann with 
admiration and interest. Surprising that Grassmann 
failed to hit off Quaternions. 

Verification of recollections about Argand. Not quite so 
enthusiastic to-day about Grassmann. Comparison 
of his own work and Grassmann's. 

Take a side, and care for nobody. Arago and the watch. 
Willing to be mentioned favourably. Skin so thick 
it can neither be pierced nor tickled. Thirtieth 
anniversary of my starting for Cambridge. 

Comparison of work with Grassmann's. Argand did not 
anticipate De Morgan. Extract from Latin poem of 
Leibnitz. 

Theorem suggested by hints from Cotes and Herschel. 

German intellect. Tobacco-smoke. Difficulties of exten- 
sion to experts used to it. D. F. Gregory. Sir W. H. 
of Edinburgh cannot abstract quantity. Leibnitz. 

Inquiries. 

Returned from fairyland, an investigation in the applica- 
tion of bi-quaternions to the inscription of polygons 
in surfaces. This day would have been birthday of 
Sister, who was a poet. Claims for himself only 
poetical temperament and feeling. 

A. DeM. April 14. Objects to his limitation of his own pretensions to poetry. 
Corrects expression about birthday. All opposite 
terms in logic come under plus and minus. Unin-r- 
sal-j particular +, &c. Recommends change of occu- 
pation from quaternions. A four-course succession. 

Illness of children. 

Table rapping. Our thoughts are read. 

Poetry runs in families more than mathematics. Lady 
Lovelace and the Differential Calculus. Verses from 
Morgan's Horologiographia, and Hylles' Arithmetic. 

Feelings after printing off a book. Anxiety about a 
word. Observations at Dunsink quinquisected by 
Royal Commissioners. Mr. Charles Thompson. 

A. DeM. July 2. A new Double Algebra. The two Sir W. H.'s, + 1 and 
- 1. Argand's lamp. Rotation of intellectual crops. 

A new symbol of operation, I*. 

Your new subject not recreation. The sailor's thive 
wishes. Differential equations. 

Book (Lectures on Quaternions] actually out. 

running suggestion of letters to be added to Sir W. H. 
H.'s name. 

Method of least squares and theory of probabilities. 
Suspects fallacy in himself and Encke. 



W. R. H. 
A. DeM. 



May 5. 
May 9. 
May 21. 



W.R. H. June 30. 



W. K. II. 
A. I 

W. K. II. 

A. I., M. 



July 7. 
July 11. 

.July Hi. 
July 18. 



W. l;. II. July 25. 



Con foils. xxvii 



From Date of Letter. 

A. DeM. July 29. Disputes * both you and Encke.' 
W.R.H. July 29, 30. Agrees with DeM. as to fallacies. 

,, Aug. 18. Suggests abridgment in De Morgan's argument (Diff. 

and Int. Calc.) as to theorem of mean value. 

Error in Moigno's Lemons de Calcul Differential. 
A. DeM. Aug. 19. Admits difficulty. Homersham Cox. The principle of 

fluxions always. gives light. 
,, Aug. 22, 23. Problem in probabilities as to set of books. Imperfect 

cerebration. 

W.R.H. Aug. 22, 24. Inverse Probabilities. The Comet. 
A. DeM. Aug. 26. Mrs. DeM.'s sight of the comet with naked eye. 

Thundering error in my Probabilities. Everyone 

makes errors in probabilities. Laplace, Poisson. 

Illustration. 
\\.K.II. Sept. 7. A theorem of Laplace. Royal visit to Dublin. The 

previous one of 1849. The first Great Exhibition 

in Dublin. Interview with the Prince Consort. 
A. DeM. Sept. 12. Presents tract on Probabilities. My thundering error. 

The theorem of Laplace. What I have not seen. 
"W. R. H. Sept. 28. British Association Meeting at Hull. Beverley Minster 

and Flamborough Cliffs. Phillips and Sedgwick. 

York. Bolton Abbey. Belfast. Carlingford. 
A. DeM. Oct. 4. Guild of Literature and Art. C. Dickens. F. Baily's 

handwriting. Lord Brougham and Quaternions. 

Conundrums. 

W.R.H. Oct. 31. Death of a friend. Guild of Literature and Art. 
A. DeM. Dec. 10. Invention of Double Algebra. 
W.R.H. Dec. 15. Telegraphic experiments in connexion with his second 

son. His cipher. Dr. Hincks. 
,, Dec. 17. Daughters. 

1854. 

A. DeM. Jan. 10. Death of his daughter. Homer's Method, and its author. 
W. R. II. Feb. 20. Homer's method. Secretary Drummond. Solution of a 

linear and partial differential equation in bi-quater- 

nions. The existence of two arbitrary functions a 

startling departure from the analogy of the older 

algebra. Carmichael. 

A. DeM. Mar. 4. Arbitrary functions queer fishes : example. 
W. R. H. Mar. 7. A transformation of the integrals involving only one 

arbitrary function. 
,, Mar. 14. Mathematical blunders of Auguste Comte. His capacity 

of admiration. 
A. DeM. Mar. 26. Comte's verbosity and self-sufficiency. His * final system 

of Knowledge.' 
,, April! 8. Dupiu's proof (regarding the lines of curvature on a 

system of orthogonal surfaces) diabolical. The 



xxviii Contents. 



From Date of Letter. 

words * synthetical ' and * analytical.' Derivation of 

name of Easter. 

W.R.H. April 24. Dupin's genius. Le bon diable. De Morgan a Hippo- 
potamus. 
A. DeM. April 29. Dupin a decent devil in mathematics. Intends to write 

letters of advice to young mathematicians, with a 

Hippopotamus on the title page. 
W. R. H. April 25. Quaternions a Calculus. 
A. DeM. April 29. Quaternions certainly a Calculus. My course of teaching. 

Infinitesimals. 
W.R.H. May 12, 20. Calculation with quaternions, geometrical interpretation 

in the background. Simplest way of proving by 

calculation the modular theorem. 
May 25. The Electro-magnetic Quaternion. (Quaternions furnish 

a CALCULUS OF POLARITIES). 
May 27. Withdraws the epithet Electro-magnetic as assuming too 

much. Dr. Lloyd. 
A. DeM. July 10. Rule for giving an extempore lecture. Stevinus. Joseph 

Fenn's ' Instructions, &c.', at the Dublin Society, 

1772, uses notation of the differential calculus. 
W. R. H. July 14. Mourey's proof that every algebraic equation has a root. 

Versors, tensors. A triplet- system. 

Aug. 11. Cipher. Asks De Morgan if he would be stakeholder. 
A. DeM. Aug. 12. Anything but decipherer. Tables of quarter squares. 

1855. The Potato. 

W.R.H. Jan. 3. Payment by Board of Printer's bill. ' Extensions of Quater- 
nions.' Would require some time to read. Easy to you. 
A. DeM. Jan. 6. A shorter demonstration is time saved in teaching. 

Example of arguing in a circle. Double algebra and 

quaternions. 
W. R. H. Jan. 9. Find I can understand my own researches on Dynamics. 

Connected this method with quaternions in is I."*. 
A. DeM Jan. 15. A remonstrance. Read an old paper of my own not 

knowing its author : pleased with it and gratified by 

this testimony to myself. 
,, April 29. A fundamental method of Newton generally unknown. 

Gives the problems. Carmichael on the Calculus of 

Operations. Who is he? Rouse out from Ilu> 

<iu;iternions. 
W. li. H. May 2, 4. Up all ni-ht with the moon. 'He' or 'It' of an infant. 

A 'young pagan. 9 Mrs. Wilde: her son Oscar. 

Her remark on the Observatory. More sociable. 

A DeM. May 16. It ' for a baby. 'Shortened.' Sponsors' undertakings. 
\\ l:. II. .Inly 11. Decimal coinage: Mr. Lowe. Maria Edgeworth's pun. 

II. i 
A. 1KM. .lul\ 18. Tun-. I'itrlmi- into a j<-ku a good service, Jokea <>n 

Ben 



Contents. xxix 



From Date of Letter. 

A. DeM. Sept. 5. Refers to his own article on Brewster's Newton. Lying 
fallow. Sheepshanks. Ballads : Curiosities in the 
history of popular ballads. Simplicity an ultimate 
attainment. Euclid I. 47. Kelland. Airy. 

\V. It. II. Oct. 2, 24. Glasgow. Jessop on Decimal System. ' Morning Regis- 
ter.' Extensions of Quaternions. 

A. DeM. Oct. 29. The * Fine Old Irish Gentleman.' Decimal Coinage : 
Lord Monteagle, Article in National Review. Mr. 
Jessop. Galileo. Anti-Romanist joke. 

W. R. H. Nov. 1. Decimal Coinage. Abduction of Miss Arbuthnot. 

A. DeM. Nov. 4. The abduction case. English legal-headedness. 

1856. 

,, Mar. 3. The words of the ' Fine Old Irish Gentleman.' Additional 

stanza by De Morgan. 

W. R. H. June 11. A protest against omission of his name from Grant's 
History of Physical Astronomy. Jacobi. Donkin. 
Cay ley. Houel. 

A. DeM. Dec. 26. Death of Sir W. Hamilton of Edinburgh. Obituary 
notice by A. DeM. The creed of the Decimal 
1857. Association. Isonosical lines. 

,, Jan. 12. Air of ' Irish Gent.' in A. DeM.'s musical notation. 

Note. Petrie on this notation. 
Jun. 18. Sir W. H. of E. on Mathematics. 

W. R. H. Mar. 22. Early days at Glasnevin. Varenii Geographia Generalis 
edited by Isaac Newton. Instance of historical 
inaccuracy on a recent and public occurrence. 

A. DeM. May 4. Newton's Varenius. Story for story. What is history ? 
Libri. 

W. R. H. May 20. Salmon and Quaternions. 

A. DeM. June 15. Salmon's name. Your son. Affirmative and negative 
as applied to propositions : definition must be 
enlarged. 

\V. It. H. June 24. The Times. Its powers of face. 

A. 1H-M. July 3. Bulk of the Times staff Irish. Newspapers mirrors. 

,, Aug. 3. Quaternions not for ladies. Approves W. R. H.'s 
Decimal Coinage Bill. Archibald Smith. Preserv- 
ing letters. Specimen of my folly in a letter to 
Mansel. Wills. Logics and Mathematics. 

W.R. H. Aug. 7. Terms with Salmon. With MacCullagh. Notion of 
writing a Tract on Cones of the Second Order treated 
by Quaternions. Asks for opinion. 

A. DeM. Aug. 14. Opinion as to such a book. 

\\Mi.II. Aug. 14, 17. Abandons the project. Notion of another tract. Letters 
to Salmon. British Association. 

A. DeM. Aug. 21. British Association. Am not in sympathy with physical 

philosophers. Object to travelling. 
,, Nov. 15. What curves cannot include a space ? No news to tell. 



XXX 



Contents. 



From Data of Letter. 

W K.I I. Nov. 16, Dec. 4. Definite Integrals and Transformation of Series. 
Theorems (A) (B) (C) (D) (E). 

A. DeM. Nov. 20. Cauchy's ' principal value.' Curves that do not inclose 
space : elementary proof . Proof that every equation 
has a root. 

W. R. H. Nov. 23. Cauchy's proof. Mourey's method. Writing merely to 
free the mind mutually allowed. Professor Stokes on 
Definite Integrals. A correction. 

A. DeM. Nov. 27. Every convex spherical polygon has its supplemental 
polygon. Cauchy's proof. Mourey's. Argand. Airy 
and <v/- 1 Herschel on a certain discord. 

W. R. H. Nov. 28. Correcting misprints^ Definite Integrals. 

A. DeM. Dec. 6. Airy's paper on \/-l. ' In all haste.' 

W. R. H. Dec. 18. G. Stokes on my Definite Integrals. 

A. DeM. Dec. 19. Tour last letters [on Definite Integrals] a memoir in 
themselves. 

W. R. H. Dec. 14. Transition from Douhle to Quadruple Algebra. 
,, Dec. 22. Herschel's wit in re Decimal Coinage. 
,, Dec. 23. A convert to Divergents. 
1858. 

A. DeM. Jan. 1. Comment on letter of Dec. 14. Vindication of Cauchy. 
The French and Divergent Series. Four sorts of 
Series. 

W. R. H. Jan. 4. Cauchy, admire his mathematical talents. My investiga- 
tions by quaternions respecting families of surfaces 
the germ of a new Calculus of Partial Differentials. 
Great importance of * alternating series.' Where 
Mourey succeeded, where he failed. 

A. DeM. Feb. 20. The logicians too mathematical. Organ-tuning by beats. 
March 1. Pascal's Theorem. Projection. Apol- 
lonius. 

W. R. H. Mar. 3. Pascal's Theorem. Moebius. 

A. DeM. Mar. 7. Projection. Pascal's Theorem the fundamental instru- 
ment. 

,, Mar. 23. The police and the Trinity College lads. A shillaly 
tournament. 

W. R. H. Mar. 30. Your construction of hyperbola derivable from Pascal's 
Theorem, which I have considered fundamental. 
Pascal above Chasles in inventive genius. 

A. DeM. April 1. Newton's equation. Chanting it. Whewell's lettn- 
1'uz/le. 

W.K.H. April 4, 0,10. Cipher. Series for/ (x + A) with a foreign and arbitrary 
constant. Silver wedding. A formula of Lnphuv 
analogous to mine. 

A. j)f.M. April 11. Anticipation of your theorem by Mti i] ill v. Anticipation 
of life. 



Contents. xxxi 



From Date of Letter. 

W. 11. H. April 14. Admits anticipation by Murphy of one of his theorems. 
A more general series. 

A. DeM. May 16. The two sides of Logic, mathematical and metaphysical. 

W. 11. H. May 26. Announces finishing and despatch of his long letter on 
Definite Integrals. 

A. DeM. June 3. Read through your 72-page letter: 'a collection of 
memoirs, a real inroad into the territory of divergent 
series.' Alternation safety in series, not convergency. 
Motto from Hobbes for my tract on Logics and 
Mathematics. Sir W. Hamilton of Edinburgh and 
4 quantity.' 

,, July 27. The two Sir William Hamiltons. Logic of the word 
4 of.' Anagrams. Newton and Huyghens. Syllo- 
gism implying relation in copula. A near relation. 
London air. Logical examples. P. M. A. C. F. 

W.R.H. Sept. 13, 14. Visit to Trim. Daughter dangerously ill. Dr. and Mrs. 
Butler. Oct. 13. My differentials what Newton 
1859. would have called fluxions. 

,, Jan. 3. Additional conclusions respecting Definite Integrals. 

Hon. Mrs. Ward. 

,, Mar. 11. Monthly Notices of Royal Astronomical Society. Reports ? 

DeM. Mar. 27. Do not write much in Reports. Main : Donati. 
,, No date of time. New date of place. Chalcot Villas. J. Smith and 
circle-squaring. Your short proof. Daughter's 
restoration to health. Advice to her. 

\V. I MI. Oct. 9. A plea for Italics and Capitals. Duty to Insurance 
Offices. 

A. DeM. Oct. 11. Insurance Offices : a distinction as to duty towards them. 
Elimination. Transition. Illustrations. 

W. R. H. Dec. 1. Deference to your criticisms. 

1860. 
A. DeM. May 26. An outcry for information as to himself. Report I shall 

circulate. Where was Sam Johnson in 1745-6 ? 
W. R. II. July 9. Touring in Wicklow. 

A. DeM. Nov. 6. Received your letter (of October 29 on Geometrical Nets 
in Space). Reviewing Sir W. H. of Edinburgh in 
Athenceum. He certainly never read Euclid I. 1. to 
understand it. C. Dickens knew it better. 

,, Dec. 18. Glad the affair (with Tait) is so well arranged. My con- 
troversy with Mansel. Very good opponents. Bad 
practice of establishing exact science by quota- 
1861. tions. 

,, April 29. Tried pleurisy. Sir W. H. and ' some at least.' 
W.R.H. May 1. Gout does not disturb my mind. TEOTA. Lady Hamilton 
wants to see A. DeM. 



XXX11 



Contents. 



Date of Letter. 

A. DeM. May 31. Mr. Smith wants more killing. Convincing Mr. Smith. 
Father 0' Flaherty's pig. A jingle. Lady Hamilton 
would be disappointed in seeing me. Another Lady 
Hamilton terribly bored by me. Mr. James Smith 
coming to London. Rhyme to Timbuctoo. Pun on 
Airy. 

My daughter's remark on my postirig in one evening 
notes to two Lady Hamiltons. * Geometrical sighs.' 
Challenge to De Morgan. A puzzle from him. 

My daughter pleasantly angry. Irish Church Canon. 

The anger will cool. Delight in defying control : example. 

Compares the Lectures with the unfinished Elements. 



W.R.H. June 30. 



\. D, M. 
W.li. II. 



Aug. 10. 
Aug. 12. 
Nov. 9. 

1862. 
Mar. 6. 
Mar. 8. 



A. DeM. Mar. 6. Inferential non -affirmation. 

W. R. H. Mar. 8. Berkeley's criticism on remarks of Newton. Newton not 
consistent in his philosophy. I stick to the/ 
quantities. Thomas Simpson. 

A. DeM. Mar. 11. Newton shuffling. My autograph for Miss Helen. 

W. U. H. Mar. 14. Have adopted Finite Differentials. 

A. DeM. Mar. 15. Differentials. Lacroix. Legendre. Limits. 

W. R. H. Mar. 18. Dispute your negative. Lagrange unreasonably objects 
to modern use of the word limit. My differentials 
not infinitesimals. Philosophy of the Infinite. 
Science of the Infinite. 

A. DeM. Mar. 23. Is differential to be used for infinite multiple of a diffe- 
rential. Supports his negative. Suggests the word 
primo-rationals instead of differentials : also terms 



W.R.H. Mar. 25. 



for Y, k, and h. 
n 



Is quasi universal usage in favour of considering a diffe- 
rential as an infinitesimal? Lacroix. Peacock. 
Cauchy. Moigno. When was * Differential ' natura- 
lised P Euler. 

A. DeM. April 1,7. Ranges the authorities on both sides of the differential 
question. Date of introduction of Diff. Calc. into 
Cambridge, x rate of y. 

W. R. H. April 19. Berkeley in earnest against fluxions as well as against 
matter. 

A. DeM. April 20. Infinitesimals before Leibnitz. Newton got hint of 
fluxions from intensionists. 

W . K. II. April 22. x rate of y won't serve me. Infinitesimals may be intro- 
duced into quaternions. The only question as to 
diflerentials. Weight of authority on your side. 

A DeM. April 'Jl. Hail'* n . -tilim ai it \ . A joke. My objeetion to your 
l-i-ai-tii'L- as to dillrri niials. //// tinl<ilt <///< /,V, 
tim. partim Lutinr. 



Contents. xxxiii 



From Date of Letter. 

\V. U. II. July 2. Differentials of quaternions. I treat them as infinitesimals 

in integration. 

A. IK'M. June -1. Change of local date in verse. 
W.K. II. Sept. l.j. Rober's inscription of a heptagon in a circle. H>rn-r'~ 

method. 

A. I H M. Sept. 18,23,29. Homer. Young. Hair-compasses. Rheticu* <>\ ' 1'iti 
cus. Cavendish. Involution and other articles in 
English Cyclopcedia. Conies or quarics. Infinity : 
twice as many pints as quarts in infinite space 't 
1863. Amosgepotically. 

,, April 3. Infinitely small quantities ? 
\V. K. H. April 16. Line at infinity in given plane. Plane and circle at 

infinity in space. 
A. DeM. April 26. Expediency of waiting. Cambridge Undergraduates. 

Infinity a light to guide. 

,, July 31. Collecting my queer books. Whe well's apothegm. 
,, Oct. 21. Boyer's contracted multiplication. Short methods don't 

pay. Mr. Weller. 
\V. R. H. Oct. 23. Those can use who don't want it. 
A. DeM. Nov. 10. Col. Oakes's rule for multiplication. 
\V. 11. H. Oct. 24. Homer's Method. Complex Evolution ? Complex Invo- 
lution. 
A. DeM. Oct. 28. Evolution and Involution. Teachers of calculation. 

James Smith. 666. Cabbala. 
W. 11. H. Nov. 10. Cotes's Harmonia Mensurarum. RATIO. A duplicate 
ratio. Briggs. Napier. Modulus. Modular Angle. 
A. DeM. Nov. 14. Cotes probably not intentionally unjust to Napier. 
Napier did not use Naperian logs. Object to Halley's 
and Cotes's Mensura rationis. Cotes and Newtori 
infinitesimalists. Newton a momentarian. 
W.R. H. Nov. 17. Napier's * Canon Mirificus.' Importance of distinguish- 
ing between Ratio and Quotient. A Quaternion, 
geometrically considered, a Quotient of Vectors, but 
do not choose to call it a Ratio of Vectors. 
A. 1> M. Dec. 31. Ratio and Quotient. Duplicate ratio. A riddle. Two 
papers. Another riddle. Shakespeare tercentenary. 
1864. Note. 

W. R. H. June 8. Logic of two Controversialists, Roman Catholic and 

Protestant. Father Tom. 
A. DeM. June 22. The two controversialists. Middle term in a syllogism. 

Anti-root. Signal-factor. 
W. R. H. June 24. The controversialists. Anti-root. Signal-factor: Argand, 
Peacock. Double Algebra interprets too much. 
Algebraic geometry requires imaginaries. Expected 
once to find in Quaternions a new set of geometrical 
imaginaries. No such instance. 
VOL. in. c 



XXXIV 



Contents. 



From 

A. I'- M. 



\\.\\. H. 
A. Dr.M. 

W.K. II. 
A. DeM. 

W. K. H. 
A. DeM, 



Date of Letter. 

In ne 26. Case in which hoth middles may be particular. Double 
algebra one of the perfect common algebras. Quater- 
nions one of the triple algebras. 

July 4. Vision. Habitually a double-seer. Binocular vision not 
necessary. Amount of obligation to touch. Visible 
distance. Do we believe we see the very trees ? 
Inly 17. Strictly unocular. Not aware of difference from a short- 
sighted person with two eyes. Obligation to touch. 
Can draw a straight line and circles. 
Your * Unocular ' a right correction of my ' monocular/ 

' Homofocal,' ' Confocal.' 

Seeing the very objects. So words for objects. Distance 
recognisable without touch. A second eye would 
add to quantity of nervous action. Illustration. 
Peroxide of hydrogen. 

Hamilton of E. on Reid. Interested in Hamilton's Philo- 
sophy. 

Hamilton of E. interesting even in Logic when quantity 
is not in question. At Oxford. Originality <>i 
thought of putting sign of quantity to the predicate. 
Agree more with Ingleby than with Hamilton as to Per- 
ception of an External Universe. Axioms ascribed 
to Euclid probably not genuine. Sir W. Hamilton'* 
view of Philosophy differs essentially from mine. 
Sept. 22. The Declaration of Scientific men. Herschel's stinging 

answer. 
Sept. 23,30. Herschel's intention in refusing to sign. Sir John 

Bowring. 

, Oct. 2. Herschel objects to Declaration as mischievous. Declara- 
tion translated into English. Illustration. Cui 
bono. Herschel not a timid man. 
Dec. 13. Pension for Professor Boole's wife and daughters. 
Influence of the Irish in the House of Commons. 
Acknowledgment of kindness. 



July ID. 
July 31. 

July 4. 
July 9. 



\V. K. H. July 9. 



A. DeM 
W.K.H. 
A. DeM 



Contents. xxxv 



APPENDIX. 

VOL, II., page 537 : 

Two i-oi'i LAI; ACCOUNTS OF Tin; (QUATERNION CALCULUS: 

PAOI 

1. From Sir W. R. Hamilton to Rev. James Hamilton, . 

2. Elementary Sketch, ........ 

VOL. II., page 543: 

MAEDLEB'S CENTBAL SUIT, ........ 037 



VOL. III., pages 199, 200: 

Additions to Contents of Elements of Quaternion*, . . . tilo 

Intended Notes A, B, and C to Elements of Quaternions^ . . 641 

Errata in Lectures on Quaternions and in Elements of Quaternion* } nil' 

VOL. III., page 200 (Note 2) : 

Memoirs in which Determinants are used, ..... (5io 

VOL. II., page 103 : 

Kant's Philosophy : Fragment of a draft of a letter to A. De Vere, 



643 



LIST OF PAI-KUS, MEMOIRS, ADDRESSES, AND BOOKS PUBLISHED BY SIR 
WILLIAM ROWAN HAMILTON, AND OF NOTICES OF COMMUNICA- 
TIONS; ALSO OP POSTHUMOUS PUBLICATIONS, AND OF BOOKS 
DEALING WITH QuATKHNIONS, ...... 045 

INDEX, 659 



COliRIGENDA. 



VOLUME III. 

THE reader is particularly requested to make the following corrections in the ^tats at the top of the 
pages specified : 

Pages 51, 83, 55, . . . for 50 read 51. 

,,83 ,,51 52. 

117,119 53 54. 

127,129,131,133, 54 55. 

135, 137, liJ9, . . 55 5G. 

141,143 55 57. 

hage 7, line '2:1, for latter read later. 

7, note, after 'supra ' insert Vol. II. 

60, line 21, /or deserters read dissenters. 

69, L 25, after integers for comma insert . 

135, 1. 14,/or Charles Mansfield read Clements Mansfield. 

379, L 17,/or Havil, Great Southampton-street, read Havil-strvet, Southampton-street. 

472, L 21 after function insert O,. 

,, 592, 1. 26,/or October read November. 

f Miimi errors are left to the intelligence ot the reader]. 



LIFE 



OF 



SIR WILLIAM ROWAN HAMILTON. 



CHAPTER XXXIX. 

THE DARGLE. LIVERPOOL MEETING. VISIT TO WESTMORLAND. 

(1854.) 

THE last preceding chapter of this work recorded the publication,, 
ten years after he had discovered the Calculus, of Sir William R. 
Hamilton's Lectures on Quaternions, and the reception by him from 
his scientific brethren of most gratifying testimonies to the value 
of the work. Both the publication and these results of it must 
have been an immense relief to Hamilton's mind and spirits. 
And indeed that relief had been greatly needed ; for, as may be 
remembered, his intellect had been long continuously tasked, and 
he had passed through a period disturbed by much care and emo- 
tion. Early in the year 1854 he received from an old friend a 
letter of thanks for a presentation copy of the book, which, though 
containing a disclaimer of the ability to read and judge of it, 
could scarcely have been less welcome to the author than homage 
from the highest mathematical authority. Its writer was Pro- 
fessor Sedgwick, the veteran geologist, who in it expresses, in his 
own natural and vigorous style, his affectionate feelings towards 
Hamilton, and in regard to himself manifests the indomitable 
spirit which, carrying on a constant battle with ill-health, was 
one of the many charms of his noble character. I here give this 
letter, and add a sequel of shortly subsequent date and equally 
characteristic. 

VOL. 111. B 



2 Life of Sir William Rowan Hamilton. [1854. 

From PROFESSOR SEDGWICK, F.R.S., to SIR W. E. HAMILTON. 

CAMBHIDGE, JfarcA 26, 1854. 

' How long your Lectures on Quaternions has been at my rooms 
is more than I can tell you: but here it is before me, with "the 
author's affectionate regards" written on the title-page. I am 
very oblivious ; so I may have thanked you before ; but I have 
no remembrance of having sent you a former letter of thanks, and 
I now do thank you with my whole heart. Your book is quite out 
of my depth, and I am far too old, and too busy in the works of 
my own craft, to think of learning to swim in the wide ocean of 
your mathematical speculations ; but I can, and do, wish you good 
speed, and I trust that your life, spent so nobly in soaring after 
truth, may be continued to you in honour and happiness for many 
years to come. I have had a miserable winter. In the October 
Term I lectured (my thirty-sixth Course) to the largest class I ever 
had at Cambridge ; and I always had a good class, even during 
the twelve or fourteen years when the professorial lectures were, 
generally, in a state of great depression. Now they are much re- 
vived. Perhaps the crowded state of my lecture-room excited me too 
much ; or perhaps I this year had a warning that I am becomiDg 
old, and unfit for my former tasks. Be this as it may, I did break 
down during the middle of my course ; and I was forced for fifteen 
days to be as mute as St. Anthony's congregation in the Gulf of 
Genoa (surely it was there or thereabouts that he preached to the 
fishes ; but you can, I dare say, eliminate the fact without the help 
of quaternions). During the said fifteen days my inner man was 
washed by streams of rattling cathartics ; and as for my poor head, 
they cured its congestion by sucking all the blood out of it by 
help of leeches applied three different times to my temples, the 
back of my ears, and the back of my neck. I then finished my 
course, though in a languid and unsatisfactory manner. The 
general reduction of my system made me unusually sensitive to 
atmospheric impressions, and before the end of Term I caught al 
oold which assumed, in a day or two, the type of bronchitis. For 
twenty-nine days I remained a prisoner in my own room, and in 
a state of miserable stupefaction. The weather then became 
miMrr, and about the middle of January (muffled to the ears, and 
with a respirator over my mouth and nose) I went to my house at 



AKTVT. 48.] Letter from Sedgwick. 3 

Norwich ; and under the care and love of a niece* I gradually 
came round, and again tried to spread my sails to the wind when 
the weather was mild and the sun was warm. I returned to Cam- 
bridge early in this month. I am still a grumbler. A remnant 
of bronchitis still clings to my chest ; suppressed gout takes away 
my sleep ; my kidneys are doing their work in a grating fashion ; 
and I have every day, especially every night, long fits of coal- 
black melancholy. A quaternion of maladies ! Do send me some 
formula by help of which I may so doctor them that they may all 
become imaginary quantities or positively equal to nothing. Any 
change will be for the better. Try some transformation or other ; 
for I am at present both stupified and demoralized. I send you 
this dissection of myself as a kind of lame apology for not having 
thanked you sooner. Your book was, I believe, on my table 
among a multitude of parcels I found there, about three weeks 
since, when I returned from Norwich. Give my best remem- 
brances to Professor Lloyd and the other members of the dear 
late Provost's family. Ever, my dear Sir "William, truly and 
gratefully yours. 

* P. S. I may well talk of getting old, for on the 22nd of this 
month I entered on my 70th year, and I have now resided very 
nearly half a century in Cambridge. I was a green Freshman in 
1804.' 

From the SAME to the SAME. 

' CAMBRIDGE, April 19, 1854. 

* Yerses I have none to send you, for I never wrote a line of 
poetry in my life at least in my own Yorkshire tongue. My master, 
under the stimulus of birch, did force out of me some Latin lines that 
would scan ; but I dare say he never had a drop of poetry from 
me. No muse ever rocked my cradle : but I love good poetry 
and good poets, and I do fervently thank you for your last letter 
and the enclosed poem. To offer you one of my geological Papers 
would be like asking you to lunch on a dish of brick dust. I have, 
however, found a copy of my letters to Wordsworth, which are a 
little more popular than my Paper read at Somerset House. I am 
not so unreasonable as to ask you to read them. I only ask you to 

* Miss Isabella Sedgwick. 
B2 



4 Life of Sir William Rowan Hamilton. [1854. 

read from page 51 ... 53, and the concluding words beginning 
page 90. 

* Pray accept this small and humble offering as a mark of my 
regard and my affectionate remembrance of your kindness. 

* Ever, my dear Sir William, truly and gratefully yours.' 

Sedgwick's 'charming letter/ as Hamilton called it, was 
forwarded, on account of the message at its close, to Professor 
Lloyd, and with it a note of De Morgan's, in his correspon- 
dence with whom he had found occasion to criticise some 
assertions of Auguste Comte's in connexion with mathematics. 
What passed between Hamilton and De Morgan on the subject 
may be found in its place at the end of this volume, but I 
insert here Hamilton's reference to it in his note to Lloyd. 

From SIR W. E. HAMILTON to PROFESSOR LLOYD, D. D. 

l 1854. 



* ... De Morgan I have, I think, not seen oftener than once, or 
say twice, in my life, but the triplets and quaternions threw us 
into a very free and familiar correspondence ; and I spoke to him 
of Comte lately with a degree of freedom, which might well have 
been called flippant, if my criticism on one part of the writings of 
so celebrated a man had been in any way designed for publicity. 
Some of the mistakes may be mere slips of the pen, but others (I 
still conceive) indicate a confusion of thought on Comte's part, on 
subjects which he professes to treat of.' 

In May Hamilton wrote an important letter to John T. 
Graves, in which he comments on Mr. Graves's theory of Octaves 
(a theory arrived at independently by Mr. Cayley), and proct 
to announce that 'one of my recent extensions of Quaternions 
gives a theory of quadrinomes, 9 and, *I have found also a system of 
ads.' A postscript refers in appreciative terms to the re- 
mrclics of th,. Key. T. 1'. Kiikman, and on this account I insert 
it in a note, leaving the main body of the letter for a place in 



AETAT. 48.] Electro- Magnetism. T. P. Kirkman. 5 

Hamilton's scientific correspondence.* The same must be the 
destination of some still more important letters to Dr. Lloyd. 
They set forth what he calls a conjecture suggested by Qua- 
ternions which might prove * a physical discovery respecting 
the mutual action of two elements of the same, or of two diffe- 
rent (electro-magnetic) currents, considered as exerting (in 
addition to Ampere's attractive or repulsive force) a certain 
direct ire force, or as producing a system of two contrary couples? 
He afterwards saw reason to doubt of the physical applicability of 
what he had called provisionally his electro-magnetic Quaternion ; 
but Lloyd continued to assign to it a high possible value in re- 
lation to the theory of Electro-magnetism. Dr. Lloyd's words 
are 



' Nay 31. I am greatly interested with your electro-dj^namic 
Quaternion. It seems to me to promise (not a new physical dis- 
covery, but what is yet more interesting) a theoretical explanation 
of the fundamental facts of electro-magnetism. . . . The similarity 
(or agreement) to these [Biot's laws, representing the action of an 
infinitely small magnet upon a magnetic particle] of the laws which 
govern your vectors, give, I think, ground for hope that you will 
be able, through it, to explain the true physical relation between 
the electric current and the magnet. And if so, the discovery will 
indeed be a great one.' 

I add here a postscript to one of these letters to Lloyd, in- 
dicating the fertility at this time of Hamilton's mathematical 
imagination : 

* 'May 16, 1854. I have little more than looked into Kirkman's Pluqua- 
ternions (Philosophical Mnt/azine, end of 1848), but am sure that they are very 
interesting, and ought to be studied, if the octaves are to be pursued. You 
know t object to his use of the word ^'quaternions, which I wish to reserve for 
<*l -f V- 1 Q', Q and Q' being quaternions. Did you ever consider whether he has 
been sufficiently cautious in his use of the association principle ? And must his 
various " imaginary duads " finally remain as such ? Are his final conclusions 
about sums of squares confirmed ? He is a very clever fellow. My son William 
Edwin has lately received another science honour in T. C. D. Salmon's Conic 
Sections he knows a good deal about, and is now entering seriously on the 
Differential Calculus, of which I must soon tell you my definition.' See 
Elements of Quaternions, pp. 99, 100. 



Life of Sir William Rowan Hamilton. [1854. 

30. P. S. This month of May has been an unusually 
original (if not an unusually husy) one with me, whatever good 
may or not have come out of this last speculation. At least four 
distinct (though not unconnected) trains of inquiry have opened 
themselves, though I may not be able to pursue any of them far 
at present ; and I could not continue the writing of a letter to 
Carmichael* this morning, which had been begun yesterday even- 
ing without perceiving a fifth line of future investigation. All 
this, however, is unfavourable to routine duty : and, besides, we 
are at present in much anxiety here about the health of my dear 
little daughter.' 

A cheer was given to Hamilton's spirits this summer by two 
events : in July the living of Loughcrew was, through his influence, 
conferred by the Lord Lieutenant, Earl De Grey, on the Rev. 
James Alexander Hamilton, son of the uncle to whom he had 
been so deeply indebted for his education ; and in August the 
Board of Trinity College relieved him of all remaining liability 
for the expense of printing the Lectures on Quaternions. They had 
previously contributed 200 towards this object, but a balance of 
100 remained due, and the prospect of having to pay this sum 
was the source to Hamilton of much anxiety. His cause was- 
warmly advocated by Dr. Luby, Dr. Todd, and Dr. Lloyd, and 
the vote in favour of the grant was unanimous. Dr. Luby 
had previously written to Hamilton, saying : ' I cannot con- 
ceive that the Board would suffer you to be at any loss by the 
publication of a work which is so noble a monument of human 
science, and which does so much honour to our University.' A 
pleasant note from Dr. Lloyd, of subsequent date, makes the 
following reference to what passed on the occasion. * I have often 

The Bev. Robert Carmichael, mentioned here, was a Fellow of Trinity 
College, who carried on original researches in the region of Quaternions, and 
corresponded with Hamilton on the subject. One formula (/> - \D, -j v ) V= O t 
connected with the celebrated equation of Laplace, Hamilton proposed to call 
4 Carmichael's Equation* : see Proceedings, R. I. A. Hamilton somewhere notes, 
concerning Carmichael, that 'he often made mistakes, but that he had the tem- 
perament of genius.' His life was out short at the early age of 31. 



AETAT. 49.] Visit to KilcroHcy. 



thought since I saw you with equal pleasure and pride of the 
Quaternion you told me of, whose unit-vectors are (not /,/, 7r, but) 
H., J., L. ;* and I made some use of it at the Board the other 
day in solving the problem of Gill's account ! I must add, however, 
that the members did not need much demonstration to arrive at the 
conclusion which they came to.' 

Dr. Lloyd at this time lived in a country house called Kilcroney, 
situated near the entrance of the Dargle, and here he and Mrs. 
Lloyd hospitably received many distinguished persons, foreign and 
native, whom the eminent qualities and accomplishments of the 
host and hostess, and the rare loveliness of the surrounding woods, 
and waters, and mountains, attracted as visitors. Towards the 
close of August, Hamilton took the thought of going to see these 
faithful friends, found them at home, and was easily persuaded to 
stay the night. 

In the following Hexameters he puts on record the incidents 
of his visit. They were sent to Mr. Thomas Disney, who, by lines- 
of his own in the same metre, had provoked Hamilton's attempt to 
repay his friend in kind. It cannot be said that they cope success- 
fully with the difficulties of the metre. His earlier poem on the 
Dargle, to which he refers in the letter prefixed to some copies of 
the latter one, printed for private circulation, may be found in the 
first volume of this work, page 147. I reprint, with the Hexa- 
meters, the prefatory letter: 

From SIR W. B. HAMILTON to THOMAS DISNEY, ESQ. 

' OBSERVATORY, September 15, 1854. 

6 You will easily believe that in allowing the following verses, 
connected chiefly with the scenery of the Dargle, to be printed, 
though not published, my highest ambition has been to give some 
moments' pleasure to a few friends and acquaintances, for whom I 
had at first designed to copy the lines myself, but found that the 
task would have occupied too much time. If those verses shall 
fall into the hands of any persons unacquainted with me, I hope 

* Hamilton, Jacobi, Lagrange; supra, p. 117. 



8 Life of Sir William Rowan Hamilton. [1854. 



that on inspection they will acquit me of the presumption 
of having aspired to rival any compositions in the same unusual 
and somewhat foreign metre, which yet I may be supposed to 
have sought for my own satisfaction to imitate. 

' You know that I have never hoped to win any reward from 
poetry, beyond an occasionally elevating relaxation from pursuits 
of a different kind, and an assistance towards preserving for my- 
self, and sometimes communicating to friends, a few records of the 
pleasurable or painful feelings of what has been, upon the whole, 
a studious and happy life. My recent attempt at hexameter com- 
position, notwithstanding its confessedly imperfect execution, has 
already enabled me to enjoy more than before (you will say, per- 
haps, by force of contrast) some passages in the " Evangeline " of 
the American poet Longfellow, not to mention here any less popu- 
larly known writers in that style. But I am so far from supposing 
that I have surmounted the (by some thought insuperable) diffi- 
culties of this kind of verse in our language, that these latest lines 
of mine do not please myself so well as even that early, and indeed 
boyish poem of my own, on a subject nearly the same, which I 
venture here to append : although, as a record of the sort above 
referred to, I have chosen in this way to preserve them. You will 
see that I have abstained from introducing the name of any living 
friend ; but have thought myself permitted to allude to a dear and 
departed lady, whose name has become already classical, from its 

urrence in the published Memoirs of her deceased and illustrious 
father.' 

'BARGLE VERSES. 

'TO AN OLD FRIEND. 

4 Thanks, old friend, for your lines, in that ancient hexameter flowing : 
Thanks for describing so well to me the adventures of that night, 
And whatsoever befell, since alone in the starlight we parted. 
I have been roaming too, and have met with some pleasant adventures ; 
Huch will I now recount : accept the imperfect requital. 

' After a morning walk to the nearest skirt of the city, 
Back by the cool canal, and the long green lane to my dwelling, 
Muring I came ; and enjoyed the distant vision of Dublin, 
Seen with the valley between, and the ship-studded ocean beyond it, 
Spires, and the haunts of men, while I stood in unbroken seclusion, 
Near that small gate which you know, where the harvest waved, ripe for tli 



AKTAT. 49.] Dargle Hexameters. 



' Then to myself I said : I'll enjoy the remainder of this day ; 
Let hooks rest for awhile, and the still iu-w hthour of writing : 
I will return to men, yet see more of nature this evening. 
Lately I have not viewed so much as the borders of Wicklow. 
There, where that hill-top appears, from the seaside rising abruptly, 
Though from my sight its base by nearer ranges be hidden, 
Lovely the scenery is, and to me was in boyhood familiar. 
And with a dweller there, from early youth I'm acquainted: 
Known o'er the world his name, with our Irish science associate, 
Yet by his friends beloved, e'en more than admired by the stranger. 
Worthy his wife to be his, but of her let these verses say only, 
Her's is the name of your wife, and that of the Daughter of Wordsworth. 
Yes, I will visit them, and enjoy the scenery near them. 

1 Easily was I persuaded, when this ray resolve was accomplished, 
And when at evening hour I stood in the friendly mansion, 
There to remain for the night, and explore on the morrow the Dargle. 
He, then, and [ set forth, on that our pleasant excursion. 
Leading a child by the hand, for whom at the font we had answered, 
To us his brother came up, who was one of my old College cronies.* 
-Stoutly we battled it out, our collegiate arena when entering ; 
Till we from rivals grew friends, and such have we ever continued. 

' So we together went on, and remembered beauty received us ; 
Which, when an ardent boy, I had sought to describe in my verses : 
But had despaired, and now, much more, must despair to describe it. 
Yet, though the air was warm, I longed for a perfecter sunlight, 
Some old effects, which I missed, to bring out on the canvas of nature. 
N>r was the wish confessed by me, at last disappointed: 
Burst forth the sun in heaven ; wood and water glittered in sunshine. 

' Leaving the lovely glen, the prospect expanded around us ; 
But it was lovely still, and only succession of beauty. 
^Xot in my friendly guide had the sense of such pleasures been blunted, 
Nor our home charms despised, from his knowledge of men and of cities. 
Venice he could describe, and its first effect on a stranger ; 
Switzerland had he searched, and many a mountain ascended : 
Yet he enjoyed, like me, the view that was open before us. 
As on green slope we lay, we touched upon manifold topics ; 
Some that referred in part to new speculations of science, 
None pursuing so far, as to trouble our tranquil enjoyment : 
Talked of the planet- spheres, and are they peopled by beings, 
Gifted, like us, with mind, and, ah ! like us, loving, but sinful ? 
And, while the sheep on the hill, at our feet, were quietly grazing, 
Spoke of the BOOK DIVINE, so rich in pastoral image, 
Lambs, and the heavy with young, and the tender care of the SHRPHRKD. 

' P, irtholomaw Lloyd, Q, C. 



io Life of Sir William Rowan Hamilton. [1854. 

Then we descended to cross to the other side of the valley. 
Boat was there none, us to ferry the murmuring rivulet over : 
Deftly we picked our steps, on the small rocks, moistened and slippery. 
And I remembered the Wharf, near Bolton Priory gliding ; 
Itroadly and stilly it flowed, hy a long line of stones intersected ; 
Pleasant passage which gave to me last autumn at sunset, 
Wishing, from hill that crowned the further bank of the river, 
< >n the gray Abbey to gaze, and to muse on the White Doe of Rylstone ; 
Kre in my wanderings I reached that Strid, so fatal and famous, 
Where the pent water boils, and the Boy of Egremond perished. 
Parting at length from my friends, I set out on my homeward journey : 
1 lere in the twilight arrived, and found your epistle awaiting. 

1 Thus have I, mingling thoughts of old times with recent adventures, 
Simple however they were, and involving nothing of wonder, 
Told you, in verses rude, and metre to me unfamiliar, 
JStory of those two days, which have passed since we last met together. 

'August Wilt, 1854.' 



The British Association met this year in Liverpool, in the- 
month of September, and Hamilton attended it, principally with a 
view of securing that it should make its second visit to Dublin in 
the year 1857. This he succeeded in arranging. I find, cut out 
from the Morning Poxt of September 30, the following report of a 
speech made by him when proposing a vote of thanks to the 
foreigners who had attended the meeting. Its allusions to the 
Abbe* Moigno and Monsieur Fouoault give it a personal interest, 
and the references to poetry, and the poetic treatment of his 
subject, are characteristic of the speaker. 

'Mv LOUD HARROWBY, and LADIES and GENTLEMKN, 
4 1 have been requested to move, that the thanks of this Associa- 
tion be given to our distinguished foreign visitors. In the first 
place, allow me to thank you for having thought that I might, 
without impropriety, be put forward on such an occasion, and for 
h.-iviug .shown thereby that you do not consider me as a foreigner. 
Since I was invited to make this motion, I have had no time 
even to read the list of our foreign visitors, and must request tlu-m 
md you to acquit me of the presumption of pretending to analyse- 
that list, or of doing anything more than state my own personal 
views. Of the members from America who may be present I shall 



ARTAT. 49.] Speech at Liverpool. 1 1 

say nothing, for in England and Ireland, and I suppose especially 
in this great port of Liverpool, we scarcely count Americans as 
foreigners. From the scientific and philosophical writings of the 
( itTinans, I hope that I have learned much in my maturer years ; 
but when I was a boy the French were my masters and teachers 
in science. In Ireland, at that time, the higher branches of ana- 
lysis went commonly by the name of French mathematics; and, in 
order to study the works of the great writers in that department, 
I was obliged to learn their language. Having never yet enjoyed 
the advantage of a visit to Paris, though I have had the honour of 
being, for the last ten years, a Corresponding Member of the Insti- 
tute of France, the scientific names of that country have appeared 
to me pretty much as abstract ideas ; but I think that when 
you look upon the eminent Frenchman at my left hand you will 
agree with me in considering that lie is no mere abstract idea, but 
a very concrete, and a very pleasant body. (This allusion to the 
fat little Abbe Moigno produced great laughter at the moment.) 
Perhaps you may have thought, any of you who may have 
happened to observe me turning over the pages of a book upon 
this platform, within the last half-hour, that I have been inatten- 
tive, negligent, and almost rude, as regarded the proceedings of 
this meeting. But when I tell you that it is a work put recently 
into my hands, by my present neighbour, the Abbe Moigno, whom 
I had known to be an abstract mathematician, but who now turns 
out to be a distinguished physicist also ; and when I add that in 
this work ("Repertoire d'Optique"), I find expressed the most 
candid, intelligent, and, I must say, generous appreciation of what 
has been done by persons in these countries, you will forgive me 
that seeming neglect. As an old member of this Association, it has 
gratified me to see noticed in this Optical work of a Frenchman an 
account of experiments in Optics, which first acquired some pub- 
licity during the first Cambridge meeting of this Association, in 
the year 1833. And if, with the name of my dear Irish friend 
"Monsieur Lloyd," the name of "Monsieur Hamilton" be coupled, 
I can scarcely be expected to feel less grateful on that account. I can 
remember the time when England was at war with France. I was 
a child then, but children quickly learn, through sympathy, the 
feelings of their seniors. And I remember that France was then, 
I do not say feared, but respected, as a formidable and gallant 



12 Life of Sir William Rowan Hamilton . [1854. 

enemy. The French were then our generous and noble foes; they 
are now our generous and noble friends. I trust indeed I feel 
assured that in making such allusions as these to events which 
now agitate the world, I do not pass beyond the limits within 
which the rules of the Association confine its members. The names 
of Whig and Tory are never heard within our halls ; those party 
differences which are, perhaps, essential to the working of a free 
State, find yet no echo here. Some may, perhaps, expect that we 
should carry out this principle yet further, and that we should 
realise in our own persons the conception of the Roman lyrist : 

" Si fractus illabatur orbis, 
Impavidum ferient ruinae." 

And our great country has stood firm, while thrones were shaken, 
and nations were reeling around it. But individuals may be ex- 
cused if they exhibit a less absolute and philosophic calm, and if 
they suffer to be perceived an ardent, and even an anxious interest, 
in the fortunes of the land to which they belong, especially in this 
Society, which, if it be an association for the advancement of 
science, is yet essentially a British Association. Permit me, for a 
moment, to make one other allusion to war, and to name another 
Frenchman. When Professor William Thomson, of Glasgow, 
was lately speaking in our Mathematical Section of potential and 
actual energy, and drawing one of his many and happy illustrations 
from the gunpowder and the cannon ball, I was reminded of some 
lines of Coleridge's translation of Schiller's play of " Wallensteiii." 
In that great play, and in that beautiful translation, the elder 
Piccolomini (Octavio) is represented, if my memory serves me 
jiright, as warning against the hazards of civil convulsion that son 
of his, Max Piccolomini, on whose character the author of the 
poem appears to have lavished all his love, and to have bathed it 
with floods of lustre. The father, I think, is represented by 
S< hillcr as saying to the son, respecting the ball when it has left 
the cannon 



'* It is no longer a dead instrument; 
It lives, a spirit passes into it." 



And I must say that when we in general, and more particularly 
when a nnall party of us, of whom I was one, were permitted 
Jately by M. Foueuult to attend the exhibit I..MS of his more recent 



AETAT. 49.] Visit to Westmorland. 13. 



experiments, the same lines of English translation of German 
pot-try again occurred to my remembrance. When I saw the little 
disc of Professor Foucault possessed with what he was pleased to 
call " un esprit de contradiction" disobedient, apparently, to its 
human former but yet essentially and throughout submissive to the 
laws which the great Creator of all things has been pleased to 
impress upon matter, I admired anew the activity of the French 
intellect ; but I looked up with even greater reverence than before 
to the Supreme Giver of all intellectual and of all higher treasures.' 

From Liverpool, with his son Archibald as his companion, he 
proceeded to the Lake Country on a visit to myself and Mrs. 
Graves, then residing at Dovenest on Windermere. During this 
visit he had the satisfaction of being several times received affec- 
tionately at Rydal Mount by the widow of Wordsworth ; he 
enjoyed at Lancrigg the hospitality of Mrs. Fletcher and her 
daughter, Lady Richardson ; and he renewed a long discontinued 
intercourse with his old friends Mr. and Mrs. Eichard Napier,* 
then sojourning near Rydal. 

A letter, written in a somewhat playful spirit, to his old friend 
Lady Campbell, discovers in Hamilton the elements which were 
deeply seated in him of a reverent admiration for woman showing 
itself in the form of a grave, old-fashioned, gallantry. I have in 
a former volumef put on record how eminently fitted was Lady 
Campbell to call forth these feelings. Not a less noble example of 
womanhood was Mrs. Fletcher of Lancrigg, the principal theme of 
this letter to Lady Campbell. No one who had ever met her could 
fail to carry away the impression that he had met in her a woman 
almost unique for the combination of majestic form and features 
with a spirit and an intellect in perfect correspondence. Enthusi- 
astic and yet dignified, strong in will and judgment, warm and 
compassionate in feeling, she was the ideal of womanly grandeur 
a queen of nature's making receiving, not claiming, universal 
homage. But I need not to set out in detail her eminent qualities 

* Supra, vol. i., p. 155. f Supra, vol. i., p. 359. 



14 f'ifc of Sir William Rowan Hamilton. [1854. 



or the influence she exerted over her friends. That has been done 
by the publication, since her death, of one of the most delightful 
of autobiographies, enriched by the reminiscences of Lady Richard- 
son, its editor, and of her sister Mrs. Davy, daughters worthy of 
their mother. The act of homage to his venerable hostess which 
Hamilton records had therefore, it may be seen, the warrant of 
-a special fitness. 

From SIR "W. R. HAMILTON to LADY CAMPBELL. 

' HOUSE OF THE REV. ROBERT GRAVES, 

' DOVENEST, NEAR WlNDERMEKE, 

Friday night, October 6, 1854. 

* Will you believe that I sat next a gentleman here, at dinner 
yesterday, who asked me if I had lately seen "Pamela Campbell"? 
a name by which you know that I never presumed to call you. 
You will think that the man must have been very impertinent : 
but will, I suppose, forgive him and me, when I tell you that he 
was Mr. Richard Napier, brother to the conqueror of Scinde, who 
is, I believe, a cousin of yours, and who is pleased to claim an 
acquaintance with me of thirty years' standing. Again, at dinner 
to-day, but at another house, your name was mentioned by a lady 
and a very old and very charming one who told me that she 
remembered Sir Guy Campbell since he was " that height " (about 
the height of the table), and that she knew he had married a 
-daughter of Lord Edward Fitzgerald. I said that another Guy, 
son of her friend, had died last year in India, and that his mother 
had scarcely yet recovered from the grief occasioned by the loss. 
She inquired whether you had any other children, and I mentioned 
Sir Edward and your daughters. She wanted then to know whether 
lln-y had inherited the beauty of their parents. I said that I had 
always heard that the married daughters, of whom alone I should 
presume to speak, had been quite "the rage," for their beauty, 
\< -., at the Irish Court, and elsewhere. It is time that I should 
t< 11 you at least the name of the lady who thus catechised me. 
She is Mrs. Flddicr of Lancrigg, and she is in her 85th year. 
She took my arm, when we were going in to dinner, and her con- 
versation was really delightful. She remembers well and vividly 
many of the remarkable characters of the latter part of the last 



RTAT. 49.] Mrs. Fletcher. William Pearson. 15 

century especially those connected with Scotland, and, still more 
precisely, with Edinburgh, where she at that time resided for 
< -\ample, Dugald Stewart. She has been well acquainted with 
Wordsworth during a long subsequent residence in this neigh- 
bourhood ; and she, and her daughters (one of whom is lady 
Richardson, wife of the Arctic traveller), have also deeply appre- 
ciated him. (I have visited alone, by moonlight, the graves in 
(xrasmere churchyard, of Wordsworth and his daughter Dora, and 
others of his family whom I remember. I have also visited his 
widow, who received me several times affectionately at E/ydal 
Mount). And yet she (Mrs. Fletcher) is quite open to fresh im- 
] sessions for instance, to the merits of Matthew Arnold's poetry. 
By-the-way, Mr. Arnold and his wife were members of our dinner- 
party. To sum up all, she (Mrs. Fletcher) has almost the finest 
<>\ <'S in the world, and when retiring in the evening, I asked leave 
to kiss her hand an action which I found was considered to be 
quite commc ilfaut, and in which my son Archy followed me. 

' I write in the bedroom which once was occupied by Mrs. 
Hemans, and the view from which is lovely. It is nearly time 
for me to go to bed, especially as I have walked several miles to- 
day. Good night, dear Lady Campbell, and believe me to remain 
your affectionate friend.' 

He also met at Dovenest a remarkable man, William Pearson 
of Borderside, Crosthwaite, and his estimable wife, born Grreenhow. 
William Pearson was a self -cultured yeoman, who at this time, 
after many years spent in a Bank at Manchester, had retired to a 
small patrimonial estate on the southern border of Westmoreland. 
Here a stranger might have been surprised to find him surrounded 
by a choice collection of books representing fully the English 
poets of all ages, and (in translation) the best Grerman authors, 
poetic and metaphysical, besides the classical prose works of his 
country and the best authorities in Natural History. Of the 
habits of birds and other native creatures around him he was a 
watchful observer, and he described them in purest English, with 
a charm that suggested no disadvantageous comparison with White 
of Selborne. Soon after his marriage, which occurred late in life, 
he spent a year in Switzerland, and Wordsworth, who had long 



1 6 Life of Sir William Rowan Hamilton. [1854. 

known and esteemed him, told me that the brief records made by 
him of noteworthy incidents of travel or natural phenomena were 
models of what a tourist, eschewing the commonplace and the super- 
fluous, should indite. Wordsworth's poems he particularly loved 
for their truth in the expression of the contemplative mind occu- 
pied with outward nature and its own higher aspirations ; he 
knew most of them by heart, and his spontaneous recitation of 
them was scarcely inferior in feeling and truthfulness to that of 
the poet himself. 

I should say more of him and of Mrs. Pearson, had not the 
latter, "by a biography,* printed for the enjoyment of his friends, 
preserved from oblivion the character and some of the writings of 
her husband. I remember that at Dovenest the Yarrow poems of 
Wordsworth, illustrated by all the ballads which prompted them, 
afforded delightful subject of discussion to him and Hamilton, 
and to others taking in it a subordinate share, and that from 
poetry the keen intellect of the rural philosopher passed on to 
metaphysics, and urged the scientific sage, not without some 
reward and satisfaction, for explanations and judgments. The 
question, then much canvassed, of the plurality of inhabited 
worlds excited between them a lively difference of opinion ; 
Hamilton, though refraining from a negative conclusion, agree- 
ing with Whewell in considering that science afforded no analogy 
in favour of the idea. 

Hamilton and his boy accompanied Mr. and Mrs. Richard 
Napier across Dunmail Raise to Keswick, and proceeded by 
Whitehaven to Belfast and Carlingford, where he was again the 
guest of Mr. and Mrs. Disney. In the middle of October he was 
at home in his Observatory, where the only exciting event of the 
closing year was a visit from the deaf and dumb boys of Cl; 
mont Institution, to whom Hamilton and his assistant showed tin* 
telescopes and some of the heavenly bodies, receiving afterwards 



Papers, Letters, and Journals of William Pearson; London: Kinilv 
Fuitliiul, 1863. 



AKTAT. 49.] Lady Campbell. 1 7 

in return what he calls * a most delightful and characteristic letter 
of thanks/ In simple terms it expressed admiration of the proofs 
of God's Almighty power, and gratitude for the bread, jam, coffee, 
and milk supplied to them by Lady Hamilton. De Morgan com- 
mented in his witty way upon this letter, writing about it, if I 
remember rightly ' These D.D.s are like other D.D.s in wisely 
linking together the good things of heaven and earth.' 

Readers who remember the terms of friendship which in early 
years subsisted between Hamilton and Lady Campbell will have 
noted with pleasure that he continued to correspond with her, and 
will be glad to learn that he still enjoyed the advantage of being 
able to confide to her his inward trials, and to speak to her freely 
of his spiritual state, receiving in return kindly sympathy and 
wise counsel. As a true friend, she had not failed earnestly to 
plead with him his own cause, when she heard of the danger that 
threatened him of being dominated by a fatal habit ; and now, at 
the beginning of this year, congratulating him on the publication 
of his Lectures, she had written ' I hear the Quaternions is a 
wonderful book, and sheds a light on Ireland. You know how I 
love to be proud of you, so I need not tell you how I rejoiced to 
hear this.' In Hamilton's letters of this period to his old friend, 
the portions which are most interesting are such as it would not be 
right to give to the public, and those of slighter texture would not 
well stand alone. I must content myself with producing notes 
referring to small books published by Lady Campbell, which 
were marked by a beautiful delicacy of touch in description and 
by justness of feeling, and with extracting one passage from a 
later letter which incidentally presents a combined retrospect 
of the external and spiritual life of the writer. 

' OBSERVATORY, May 17, 1854. 

'So while you were seeking to draw me out, on May-day last, to 
talk too much about what you were pleased to call my " wonderful' * 
book, you did not give me the least hint that there was a chance 
VOL. in. c 



1 8 Life of Sir William Rowan Hamilton. [1854. 

of my so soon seeing your "charming" one . . . Now I won't 
begin to praise you even as an author, and can't say that I find 
that anyone here is disposed to wonder at your producing a de- 
lightful book. But I may mention or confess that, living so much 
out of the world as I do, it was not till yesterday that I heard of 
the publication of your Cabin by the Wayside, and forthwith 
pounced upon it at M'Grlashan's . . . 

But your book fairly carried the day, or the evening, as I was 
walking back along the Canal yesterday, after having walked 
to Dublin some hours earlier by the same path. A couple of mile- 
stones gave me very pleasant opportunities for sitting down to 
read more comfortably, and when I got home I was so selfish as 
to finish the book quite alone. But this morning I showed it to 
Lady Hamilton. ... At all events she never stopped till she had 
read aloud the whole to my little daughter, and to a son, who pre- 
tended to be reading Homer. I have put my daughter's name in 
the copy.' 

' May 18th. Unquestionably I shall accept with pleasure, not 
to say with pride, the copy which you promised me of The Story of 
an Apple; and what's more, that I may have the pleasure of reading 
it first, as a gift from you, I shall abstain from ordering the book. 
My son Archy has confessed that he was listening while his 
mother was reading your book aloud.' 

* June 15th. It was Jellett's turn, as one of the Examiners 
for Bishop Law's Mathematical Premiums, to give the annual 
dinner. I sat between him and Mr. Walsh, a clever barrister,* 
and remarked to the latter that this was the 28th time of my 
assisting at an Examination Dinner of this particular kind. 
" What a frightful restrospect ! " exclaimed he. " Do you mean 
the dinners? " I asked him. " No," said he, " but the years." I 
let the conversation on that subject drop ; but cannot altogether 
regret that I have lived so long, though I have had impatient fits. 
Or rather, in my serious moods, I feel it to be a subject of deep j 
thankfulness that so much time has been allowed for schooling a < 
process which as yet has been very imperfectly performed in my 
own case ... if I find and transcribe them [some notes written at 

Afterwards Master of the Holla. 



AETAT. 49.] Retrospect. 1 9 

the commencement of the preceding year], you will see that, when 
thoughtfully considering the question, I have heretofore recog- 
nised, as I still do recognise, a kindness in life being spared. . . . 

P. S. I have played a few games of draughts this morning 
with my daughter dear, sweet, patient child. She has very few 
sources of amusement, but she enjoys flowers, the garden, and the 
Bible.' 



C2 



2O Life of Sir William Rowan Hamilton. [1855, 



CHAPTER XL. 

CORRESPONDENCE WITH AUBREY DE VERB I WITH THE NICHOLS : 
CIPHER: MRS. WILDE. 

(1855.) 

THE year 1855 was not remarkable in Hamilton's history, either 
for scientific work or for events of importance ; yet there remains 
of it extensive record in the form of correspondence. A summary j 
of the facts, followed by some extracts from the correspondence, 
may here suffice. 

The correspondence between Hamilton and Aubrey De Vere | 
springs again into activity. It had been interrupted by an illness j 
of the latter from scarlatina which precluded for a time intercourse of k 
any kind. In its course it went on continuously into the succeed- 
ing year Mr. De Vere wrote some most interesting letters, giving , 
an account of his gradual change in religious opinion, until his j 
final submission to the Eoman Catholic Church. Hamilton declines I 
to discuss the great question controversially, but gives his friend j 
clearly to understand that his own opinions, instead of moving in j 
the same direction, had become more Protestant. Had Hamilton's I 
letters met those of his friend by argument, I should have thought I 
it a duty to print both sides of the correspondence in full; butl 
besides the limits of space which I am obliged to observe, I have toH 
consider that these letters of Mr. De Vere belong more properly to 
the history of his own life. It is to be desired, I think, that they 
may see the light, for undoubtedly they deserve meditation andjl 
discussion, as worthily expressing the views which proved d<visi\v 
with a singularly pun; and thoughtful mind. I say this, though 
myself farther removed, perhaps, from his stand-point than wafil ' 



AETAT. so.] Aubrey De Vere. The Nichols. 2 1 

Hamilton, of whom it is only fair to add that, writing subsequently 
to another friend, he calls Aubrey De Vere a * splendid type of 
llomauism.' It is true also that, although the two friends diverged 
widely as to the Church question, they continued to the end to be 
in harmony in respect to their views of religious philosophy; 
Hamilton, as well as De Vere, maintaining that, from the constitu- 
tion of the human mind, Faith had in Religion a function at least 
co-ordinate with that of Knowledge. 

The meeting of the British Association attracted Hamilton, in 
the September of this year, to Glasgow, and there, at the Obser- 
vatory of the University, he became the guest of his friend Dr. 
John Pringle Nichol; and an intimacy thus arose between him 
and the family of the accomplished astronomer, which continued 
to be highly prized on both sides. Dr. Nichol's son, now Pro- 
fessor of English Literature in his father's University, was then a 
member of Balliol College, Oxford, and already manifested that 
taste for modern literature, of which he has since given to the 
public proofs which mark him out as not only a vigorous critic, 
but a man of poetic power. The confidence which his father had 
won from Hamilton was now, with full expansion of feeling, 
imparted to the son ; and the daughter and sister of these men, 
Agnes Nichol (afterwards to become wife of Dr. Jack, Professor 
of Mathematics), herself imbued in literature, and exercising in it 
her inventive faculty, was admitted a member of the band of 
friends. The correspondence which resulted was frequent and 
animated ; but as on Hamilton's side it was mostly occupied with 
the incidents of the past, a region which the reader has already 
traversed, I refrain from quoting it, except in brief extract.* One 

* It was about this time that John Nichol, Jun., addressed to Hamilton 
two sonnets, which tended to cement the friendship between them. These 
Bonnets dwelt upon the combination in Hamilton of simplicity and affection 
with high scientific qualities ; but some imperfections of expression rendered 
their author dissatisfied with them, and he afterwards condensed their meaning 
into a single sonnet, which has appeared in the collection of poems recent ly 
published by him. A similar tribute to the memory of his father may be found 
in the same volume, and in this connexion will interest the reader. 



22 Life of Sir William Rowan Hamilton. [1855, 

subject of Hamilton's letters to Dr. Nichol and his son was a 
cipher of his own invention. Some of the extracts which follow 
will inform the reader of the great merits of this cipher, the 
almost unlimited power of variation which it admits of, and the 
practical impossibility of discovering its key. Hamilton was wont 
to say that he believed a challenge to the whole world to discover 
it would not meet a reply. 

The communication which Hamilton made to the Mathemati- 
cal Section of the British Association at Glasgow was a Paper On 
the Conception of the Anharmonic Quaternion and on its Application 
to a Theory of Involution in Space. During the week of the meet- 
ing he joined an excursion-party to the Isle of Arran, and with a 
select number of its members was received by the Duke of Hamil- 
ton at a dtjeuner at Brodick Castle, Sir Eoderick Murchison 
introducing him personally, and bespeaking for him the place 
of honour among the guests. This visit was one of great enjoy- 
ment to Hamilton. The conversation which passed led to a short 
but cordial correspondence. Hamilton had mentioned, perhaps 
quoted, to his host Aubrey De Vere's poem to the memory of the 
Highland Mary of Burns, and the Duke had the tact to recognise 
in Hamilton a namesake as well as a savant. Accordingly Hamil- 
ton afterwards forwarded to him a copy of Mr. De Vere's poem, 
adding (let it be confessed) his own Dargle Verses, and mad& 
good his title to his name by using and calling attention to the 
impression on the envelope of his father's seal, which bore the 
Hamilton crest and motto ; and in reference to his own branch of 
the family, in a subsequent letter to the Duke, he took occasion to- 
write : * Sir W. R. H. has, with the exception of his two sons 
William and Archibald, only one living male relative of his own 
name with any traceable degree of connexion : namely, his cousin 
the Eev. James Hamilton, who has the living of Loughcrew in 
Meath, and who (according to family tradition) is the heir t 
a dormant baronetcy.' 

On the day he left the Observatory of Glasgow Hamilton com* 
posed a sonnet prompted by a photograph of his daughter, which 



AKTAT. 50.] His Daughter. 23 

he had received as a gift from her on the eve of his departure from 
home. This sweet and interesting child had wound all her tendrils 
round her father's heart. In one of his letters I find him 
writing : " I think Lady Hamilton would forgive me for saying 
that the being I now love most in the world is my little daughter;" 
and elsewhere he declares that he is in the habit of judging the 
happiness of other men by considering whether they have or have 
not a daughter. It may be added, that even at this time si 10 was 
now only fifteen she had evinced much love for poetry, with 
which she had largely stored her memory, and that having been 
brought through more than one serious illness, she had been very 
recently a sufferer from an accidental burn of considerable severity. 

1 SONNET TO MY DAUGHTER. 

1 Dear patient child ! upon a bed of pain 
So lately lying, watched by tender eyes, 
Thy sun-limned face and form I dearly prize, 
Thy gift at parting ; and can see again 
Thy head bowed meekly o'er some poet-strain, 
In book out- spread, or some diviner page, 
Such as would oft thy maiden thoughts engage 
Ere yet I left thee for the ocean-plain. 
Not all the wealth of mind, not social joy, 
When Scottish Science met in converse free 
With men of other lauds, and welcomed;me, 
A. way from thee could all my heart employ. 
In starry tower, or on the sunny water, 
I blessed my loving and beloved daughter. 

4 OBSEKVATORY, GLASGOW, September 24, 1855.' 

Late in the autumn of this year a visit was paid to his own 
Observatory, which stirred into activity remembrances which had 
not ceased to be cherished by Hamilton, and enabled him to show 
the interest of a friend in persons whom he had not before seen. 
One day in October, Mr. Robert O'Brien arrived at Dunsink 
with his two young daughters and their two brothers. Mrs. 
O'Brien, formerly Ellen DeVere, had desired that her husband 
and children should be known to Hamilton and his wife, and 



24 Life of Sir William Rowan Hamilton. [I8io. 

advantage was taken of their being for a short time in Dublin to 
make an arrangement for the purpose. Friendly feeling was 
manifested on all sides, and Hamilton took special delight in 
showing to the daughters an experiment which he remembered 
having in the same meridian room shown to the mother, and never 
in the long interval to anyone else that of a key floating on 
mercury and he was prompted, when the visit was over, to com- 
pose a sonnet addressed to the two young girls, which in affec- 
tionate terms expressed the pleased and tender feelings with which 
he had welcomed them to his house, and joined them to the com- 
pany of friends, living and dead, whom he liked to associate with 
the thoughts of his garden : 

' Mary and Alice ! though before that day 
Unmet, when, through this garden wandering, 
Lately ye trod where Wordsworth once deigned sing 
Some grand old strain for me, or newer lay 
Murmured, half musing, or in hoy hood's May 
With Brinkley spent I sweet hut solemn hours, 
Listening his lovely lore of stars or flowers, 
The thought of you not soon shall pass away ! 
Your images shall rise before me here, 
With thoughts of other friends of long ago : 
Time takes not all things in his ceaseless flow, 
Fixes and hallows some : and many a year, 
Since dear ones linked with you I first did know, 
Gives me a poet's right to call you dear. 

* OBSERVATORY, November 5th, 1855.' 

In the spring of the year Hamilton met, for the first tinu-, 
Mrs. Wilde, afterwards Lady Wilde, the wife of Sir William E. 
Wilde, the eminent physician and Irish archaeologist, herself 
remarkable as a woman of warm feelings and literary faculty, 
shown in patriotic and eloquent contributions to newspapers and 
other less ephemeral periodicals, and in poems published under the 
name of Hjn-riinza 9 and, it may be truly added, as a woman of high 
aspirations and real genius. The acquaintance rapidly ripened into 
friendship, and a r-TesponJ .~, ensued in which Hamilton some- 
* n'-ted as critic of a poem by 5^m//o/, at other times coufidfd 



AETAT. 50.] Lady Wilde. Hipfxirchus. 25 

to her the story of his life. The same reasons which have led me 
to refrain from presenting in print Hamilton's correspondence 
with the Nichol family dictate a similar course in this instance. 
But it would be wrong to leave unrecorded the fact that Hamilton 
was not only interested in the mind of this gifted countrywoman, 
but esteemed highly her whole nature, in which he recognised 
many features of native nobility. 

It was at the end of 1855 that Hamilton was led by his study 
of the Almagest of Ptolemy to engage in computations connected 
with the Problem of Ilipparc/tits, which set forth the remarkable 
accuracy of the observations and deductions of Hipparchus; a 
degree of accuracy declined from, rather than improved upon, by 
his admirer and commentator of 300 years later date, Claudius 
Ptolemy. A specimen of this exercise of Hamilton's powers in 
exhibition of the work of these Fathers of Astronomical Science 
may be found in Hermathena for 1883. By his study of this 
astronomical problem Hamilton was led on to investigate, in an 
extended treatise, a geodetical problem, having much affinity, 
which had been proposed and solved by the celebrated Snellius in 
1617. This treatise remains unpublished. Among Hamilton's 
scientific correspondents during this year were, besides De Morgan, 
Lloyd, Boole, Cayley, Carmichael, Eomney Robinson. I find 
four pages (61-64), but no more, of a letter to Mr. Townsend, 
F.T.C.D., on Co-planar (or Homosphaeric) Involution. 

An extract from a letter to Mrs. Wilde, with which I intro- 
duce the non-scientific correspondence of this year, has much bio- 
graphical and ethical value, as showing that though Hamilton 
loved praise he also feared it, and that his was a mind which, with 
regard to politics, was no holder of stereotyped opinions, but 
capable of growth, and of comprehensive sympathy. After somo 
remarks (prompted by a letter from his enthusiastic correspondent) 
on the dangers of the reciprocal admiration of friends its liability 
to degenerate into reciprocal flattery Hamilton observes : 



26 Life of Sir William Rowan Hamilton. [1855. 

From SIR W. E. HAMILTON to MRS. WILDE. 

' OBSERVATORY, June 30. 

* ... At all events you are no chaplain of mine, but only a 
fair compatriot, whose Irish feelings prompt you to take the most 
favourable view in your power of one who is essentially an Irish- 
man by birth, and life, and labour, though educated by a clergy- 
man who held the ascendancy principles (from which, by very 
slow degrees, I have been through life gradually emancipating 
myself), and who would have regarded repeal as rebellion. It was 
English history, not Irish which I was taught ; and my heart still 
throbs with sympathy for that great British Empire to which, 
from childhood, I have been accustomed to consider myself as 
belonging as to my country though Ireland, as Ireland, has 
always been the object of my love and, I think you will admit, 
of my exertions. God forbid, then, that I should even smile, and 
I pray you also not to think it, at any grief, or sadness, or depres- 
sion which any feeling of patriotism may have caused you. Edu- 
cation has much to do with the direction of such a feeling, but I 
know what it is, without hope or fear of any private gain or loss, to 
sympathise with a nation. I was almost literally sick with sorrow 
at hearing of the disasters in Cabool and the Khyber Pass several 
years ago.' 

Passages from a letter written in the spring of 1885, to his old 
and kind friend, Lady Dunraven, will, by the mention of the 
sister of Aubrey De Vere, lead on the reader to the letters which 
passed in this year between him and Hamilton. 

From SIR W. E. HAMILTON to the COUNTESS DOWAGER OF 

DUNRAVEN. 

[FROM A COPY.] 

' OBSEKYATORY, May 19, 1855. 

' How often have I poured out to you in years now long past 
the secrets of my heart, and how kindly you suffered mo to do so, 
uud responded to, and comforted me ! 



AKTAT. 49.] Letter to Lady Dunravcn. 27 

" Lady, who with a mother's tenderness, 
And fond indulgent patience, nursingly 
Cherished that hope in its frail infancy. . . ."* 

* You see that I quote from memory [alluding to a verbal cor- 
rection made in his writing of the quotation], though I don't forget 
your most kindly returning me some years ago the original, such 
as it was, of those that were at least heartfelt lines. 

1 A. very unexpected and yet very natural circumstance, or 
train of circumstances, has brought back Ellen De Yere, and of 
course you, dear Lady Dunraven, with unusual freshness before 
my mind and heart, within the last few days, although you and 
she had never been forgotten. 

* My dear sister Eliza, the poetess, whom I am ashamed to 
confess that I forget whether you ever saw, but with whom Lady 
Anna Maria exchanged some friendly notes, died in my arms in 
May, 1851, having every comfort, spiritual and temporal, which it 
was possible to procure for her, and with that last poetic satis- 
faction of the evening sun shining beautifully and gloriously, 
us well as comfortingly, in for the few minutes which were her 
last. We read of the Sun of Eighteousness, and the early Chris- 
tians accepted the title of Sunday from the heathens as associated 
with Him who, on that day, rose from the dead, and triumphed 
for Himself and for us. 

* My sister Eliza left me the most entire control over her 
papers, which were many, with power to preserve or destroy. To 
some extent indeed to a large one I have used the latter power, 
by putting into the names, after reading them, a great number of 
the sheets of a journal which, I am convinced, would have brought 
me in some hundreds of pounds if I could have borne to publish 
it, but which was written for herself alone, though in her dying 
hours she gave me permission to read it, and which described 
too freely, as I judged, though without any particle of malice, our 
many visitors of long ago to this Observatory. 

4 But among the numerous papers which thus came into my 
hands, it is only very recently, after the expiration of four years 
of mourning for my sister, that I have taken courage to open what 
she considered her pet box of letters, and had confided unreservedly 

* Yol. i., p. 510. 



28 Life of Sir William Rowan Hamilton. [1855. 

to me. If I tell you that I found in that box a large number of 
letters of my own, you will make allowance for the affection which 
we felt towards each other, and which was more like that of twins 
than of a brother and sister whose ages differed by more than a 
year, but who " in early childhood, almost infancy, had wandered 
forth together fancy- fraught."* We loved each other to the last, 
and I had the satisfaction of being . . . useful to her. Indeed 
I have burned a great number of letters of hers to me, because I 
did not like to preserve records of the gratitude of one who de- 
served //, and more than all, which it was in my power to do. 

* But among the papers of my sister I have found many letters 
which had not been written by myself ; and among those I must 
confess that I have found, and even read, though I have not yet 
had courage to read through all of them, several letters from Ellen 
De Vere, and from Mrs. Robert O'Brien. At this great distance 
of time it did not seem to me an unpardonable sin, nor even any 
want of delicacy, to do so ; and I was comforted on that score by 
finding in one of the most beautiful, but also most secret, letters of 
the set the remark : " I should be glad that no person out of your 

family should know of this letter." My sister, no doubt, feared to 
agitate me by showing me at the time that very charming letter ; 
but surely I may accept, as an anticipatory forgiveness for reading 
it, the sentence which I have quoted. 

* We are told from the pulpit, and cannot, perhaps, be told too 
often, of the faults and corruptions of our nature. Most fully, 
most heartily, do I grant all that from the experience of myself. 
But I must say that my opinion of the good part of human nature, 
and especially of the nature of young ladies, has been very 
decidedly exalted by the perusal of my sister's correspondence, not 
quite completed yet. How much it struck, how much it affected 
me, to compare the ardent yet discriminative enthusiasm with 
which Ellen De Vere and Dora Wordsworth (whose grave I kissod 
by moonlight at Grasmere last autumn) wrote to my sister sepa- 
rately of their first meeting ! ' 



Vol. i., p. 495. 



AKTAT. 49.] Aubrey De Vere on Old Times. 29- 

From AUBREY DE VERB to SIR W. R. HAMILTON. 

' KILLARNEY, July 25, 1855. 

4 ... I cannot say how much I am obliged to you for them 
[letters of Sir W. R. Hamilton forwarded from the Seven 
Churches]. They are convincing proofs that you have not for- 
gotten old times. To me old times seem rather to approach one 
nearer, as one advances in life, than to recede into the distance. 
To Time may he said, in its degree, what "Wordsworth says to a 
mightier Power 

" Thou takest not away, Death." 

The present it is, not the past, that looks like a vision, except so 
far as duty and sorrow convert it into a reality. 

' The extracts which you sent me [from old letters of E. De 
Vere about Curragh] have for me a deep and touching interest, 
partly of a different kind from that which they have for you. At 
first I hardly understood what they referred to, or by whom they 
were written. Gradually the mist cleared away, and scenes which 
belonged to my boyhood, but which were the most touching I have 
ever known, came back to me with the strange pathos of the past. 
The very features of the beautiful face, referred to in the letter to 
your sister, seemed to look forth again from the long darkness 
not, however, in the radiance of youthful life, but in the yet more 
beautiful stillness of death flower-crowned, and with a sweetness 
that seemed cast down from the region of Immortality.* 

* I should wish to hear something more of your sister Eliza. 
She died, you tell me, in your arms, and four years ago. Had 
she been long ill ? I trust that her death was one of peace and 
joy, for I believe that there is sometimes a joy as well as peace in 
death that life knows not of. This, however, is granted or not 
granted, as Grod sees good for each several person, and no infe- 
rence is to be drawn from its absence. I shall be very grateful for 
a copy of your sister's poetry. There are poems of hers which can 
hardly be forgotten. Those which you tell me my sister singled 
out for special approbation were, I think, my favourites also, but 
the Columbus I liked the best of all. Some of the lines of it 

* This passage refers to a sister of the writer of the letter, who died at the 
age of fourteen. 



30 Life of Sir William Rowan Hamilton. [1855. 

recurred frequently to my memory during that long and trying 
period of change (progress I should rather call it) in my religious 
convictions, which ended in a change of my religious position a 
period in which one seemed to navigate a trackless deep, over 
which Faith alone could be one's guide (though many a " floating 
wreath impearled" gave good omen), till were heard the voices 

" Deep-murmuring ' land' in music proud." * 

* She must have left many poems unpublished. I have directed 
my last volume to be sent to you at the address you name. I 
should have wished to dedicate the volume to you, as some of the 
poems in it were included in the volume originally so dedicated. 
It would not have been suitable to do so, however, as several of 
the poems at the end of the volume express opinions with which I 
oould hardly expect you to be in sympathy. The present arrange- 
ment of the poems is, however, but temporary, and those that 
originally belonged to you will, I trust, find their way back to you 
ere long in a new edition. . . . 

* To return, however, from Poetry to Life .... I am very 
glad that you made a pilgrimage to see Mrs. Wordsworth last 
year. She is the most venerable woman in England. I visited 
her also since her widowhood. She always epoke in the most 
-cordial terms of you, as did the great Bard. He used to say you 
were the only man who had ever reminded him of Coleridge. 
You were sorry, I am sure, for poor Hartley Coleridge. His 
sister, Sara Coleridge, who died some years ago, was one of 
my most intimate friends. She inherited no small part of her 
father's genius. Do you ever read his works now ? Wordsworth 
Kleins to me to have much more accomplished his mission. . . . 
It has been a real gratification to me to find that your thoughts 
have been reverting so much of late to that Past in which my own 
find so commonly their home. The modern world, ever rushing 
into the future, is, I suppose, against all reminiscences on account 
of tlu-ir "inutflity:" but surely this is but in accordance with its 
usual materialism. "A painless memory of pain" will probably 
be part of the inheritance of the Blessed hereafter ; and we are 
surely the better for anticipating here what, we hope, awaits us in 

Columbus, by E. M. H., see vol. ii., p. 698. 



AKTAT. 50.] Hamilton and Aubrey De Vere. 31 



a happier sphere. All your recollections of my sister give me the 
same sort of satisfaction which they would give to her also, as pro- 
serving old kindness, without the alloy of old suffering though 
suffering is less to be called the alloy than the fire which clears 
the metal from the alloy. It seems to bring her back to me as she 
was when a girl : 

" There's something comes to us in lii'e, 
But more is taken quite away." 

4 It is as a girl that I chiefly like to think of her when Poetry 
and Thought seemed to give her wings. She has been happy as 
;i wife and mother ; though life has brought its cares to her as to 
all who root themselves in it ; yet I do not doubt that she would 
have been not less happy had she continued to float over the sur- 
face of life, barely touching it, as of old. Be assured that the 
feelings she expressed in her letter to your sister in 1838 are those 
she feels now. . . . 

4 As for the Extracts which I have been reading again since 
I began this letter it is a deep satisfaction to me to have them. 
They come to me like a strange, and yet near and thrilling voice, 
from the region which we call the Past, but which is past only in 
semblance, for whatever has been lives on in its influences, and is 
destined to live again.' 

From SIR "W. R. HAMILTON to AUBREY DE VERE. 

* OBSERVATORY, Auyust 4, 1855. 

4 As to our religious views, you rightly judge that we differ 
more than we once did. Even if you had stood still, which was 
(as I suppose) impossible, at the point of Catholicism which you 
occupied when we used to have the most frequent opportunities of 
conversing freely on such subjects, and when there was, I think, 
very little difference of opinion, if any, between us, yet I have not 
stood still. Yours has been, as I most fully admit, and feel, with 
respect to yourself, a vital progress, a real psychological develop- 
ment ; you are now more consistent with your own instincts, and 
so far subjectively truer. On the other hand, if you knew my 
mental history, you would be apt to say, or if too polite to say 
it, you would think that I have been retrograding, for at least 
ten years, since about the time of Mr. Montgomery's change of 



32 Life of Sir William Rowan Hamilton. [1855. 

profession. The question of progress or retrogression, as regards 
myself, I do not choose to discuss : but the (mentally) historical 
fact, expressed in words which we may both adopt, is that while 
you have been becoming, or considering yourself to become, more 
Catholic, and certainly more Roman, / have been growing gradu- 
ally more Protestant, and (as you are welcome to call it) more 
Anglican, than I once had been. 

' We are therefore, with whatever regret it may be admitted 
on both sides, by a twofold motion, less in harmony now with each 
other than we were in former years, on the subject of our religious 
feelings, impressions, and convictions that is to say, on the most 
important subjects in the world and I see no probability of this 
state of things being changed. I suppose that in saying all this 
I hardly give you any information none certainly for which your 
letters alone would not show me that you were already abundantly 
prepared. 

* If then it be painfully evident to both, that under such cir- 
cumstances there CANNOT (whatever we may both desire) be now, 
in the nature of things, or of minds, the same degree of intimacy 
between us as of old; since we could no longer talk with the same de- 
gree of unreserve on every subject which happened to present itself, 
but mmt, from the simplest instincts of courtesy, be each on his 
guard not to say what might be offensive, or at least painful to 
the other : yet WE were once so intimate, and retain still, and, as I 
trust, shall always retain, so much of regard and esteem and ap- 
preciation for each other, made tender by so many associations of 
my early youth and your boyhood, which can never be forgotten 
by either of us, that (as times go) two or three eery respectable 
i mi Menu's might easily be carved out from the fragments of 
our former and ever-to-be-remembered intimacy! It would be no 

iteration to quote the words " Heu quanto minus est cum 
reliquis versari, quam tui meminisse ! " . . . 

* I have been very much gratified, indeed, by the kind me- 
which I have received through you this morning, from one of the 
two ladies whom I have loved the best in the world and of whom 
it is a deep satisfaction to me to know, that one is happily dead, 
and the other happily living namely, the objects of my very 
early, and of my later, but still youthful poems. 

* That I should have teen the one before her death, and hu\v 



AKTAT. 50.] Close of an Era. 33 

heard to-day through you, at least by a message, from the other, 
is a matter of quiet joy, and indeed of thankfulness. It seems to 
say to me : " Your past life is rounded off ; you have lived for 
half a century, and the lad day of it has been made happy by a 
recognition from one whom you had so very long and so very well 
remembered. Let that suffice for the past, and try whether you 
cannot, for the future, not indeed exert yourself more vigorously 
(that were unreasonable and hopeless to expect), but at least work 
more calmly, suffering all agitations about former times on earth 
to be absorbed in thought of the future eternity." Decidedly 
I feel it as a close to an era, although a very consoling close to me, 
that on a day which to my own imagination, or perhaps fancy 
(for I am accustomed to have associations with, and to receive im- 
pressions from, days that seem to me remarkable), presents itself so- 
much as one of transition, as that of my reaching the age of fifty, 
at the same time that my second son, Archibald, attains exactly 
that of twenty years old, I should receive through you a kind 
message of thanks from your sister, from whom I had not heard 
in any way for so very long a time in fact, since 1831. . . .' 

From AUBREY DE VERB to SIR W. K. HAMILTON. 

'August 9, 1855. 

{ . . . These are the things that show us the marvellous depth 
and strength of the human affections : when we call to mind how 
commonly they are left without response in this world, or even if 
responded to, are opposed by invincible obstacles, we cannot resist 
the conclusion that our whole moral being finds here below but a 
rehearsal of that great part assigned to it hereafter. On earth 
there seems no proportion between the strength of the affections 
and the objects on which, or circumstances under which, they are 

rcised. Were our life an animal life, man would indeed be 
the most ill-assorted and incomplete of all animals. But this very 
incompleteness of our destinies, considered as creatures of time, 
takes its place among the many proofs that we are made for 
Eternity; and doubtless our human affections burn with a fire 
some portion of which is derived from that Love which is to 
last for ever, and to be bestowed on no human object, except as 
known and enjoyed in and through God.' 

VOL. III. D 



34 Life of Sir William Rowan Hamilton. [1855. 

From the SAME to the SAME. 

4 DINGLE, September 1, 1855. 

* ... I do not like, however, to let another post go out with- 
out sending you at least my most cordial thanks for a confidence 
which is in itself the best possible proof of the sincerity and 
fidelity of your friendship. It also assures me (and this is a 
subject of great gratification to me) that, seldom as we have met 
of late years, you do not distrust the permanence on my part of a 
friendship which I have always regarded as one of the chief 
blessings of my life. "When we met first, it was at a time when 
I had formed few friendships ; and for this reason, as well as 
others, I naturally grew into the habit of associating you with all 
those studies and aspirations which belong to growing youth, but 
do not desert our maturer life. Later experience has never tended 
to withdraw my thoughts and affections from what belonged to 
that early time ; and even to the most intimate of my later 
friends I have often been tempted to say, as Charles Lamb does 
in one of his poems, " Why did you not belong to the old, well- 
remembered times ?" . . . 

' I will take care to communicate to my sister, when I see her 
next, what you so kindly, and with such affectionate and generous 
remembrance, say respecting her; and in the meantime I may 
safely assure you that she has never forgotten you, or remembered 
you with other feelings than those of the warmest, most grateful, 
and most admiring friendship/ 

From SIR W. E. HAMILTON to AUBREY DE VERB. 

' OBSERVATORY, September 3, 1855, 
' Monday Night. 

1 So you wrote those lines to Burns's Highland Mary, which I 
had so very deeply admired, without having the slightest hope 
that I should ever know so much as the name of the author ! 
Surely they were not printed in either of your two former 
volumes. I think it was in a number of Blackwood's Magazine I 
that I first saw them, and very much they struck me ; but I had 
not the least suspicion that they were by you. Perhaps such lines I 

M 

" Honour to Scotland and to Burns " 

ought to have made me suspect that they came from a fc 



AETAT. 50.] Poems of Aubrey De Vere. 35 

source foreign, I mean, to Scotland ; but I must own that I 
-always attributed thejjpoem to some native Scotchman. I congra- 
tulate you most heartily upon the authorship of it. To anyone 
whatever it would, in* my opinion, be a fit subject of congratula- 
tion. . . . 

[Here followed critical remarks on several of the poems con- 
tained in a volume published by Aubrey De Yere in 1855.] 

* I must once more congratulate you, not merely on the 
poetical skill, but also] on what seems to me the perfect taste with 
which you have expressed your new views, and the complete 
absence of bitterness, so far as I have yet observed, even in 
manner. The change seems to have been no change to you 
(though I can well believe your statement that it cost you a 
struggle). The old poems which you have retained appear to be 
quite at home among the new ones. Yet I think that, if you look 
back upon that volume which you dedicated to me (and of course 
you could not have properly paid me the same compliment again], 
you will understand that / may have almost entirely sympathised 
with you then, and yet be separated importantly now; as, for 
instance, with regard to the Blessed Virgin Mary, whom certainly 
I am not conscious of honouring 'less than I formerly did, but 
whom you now honour otherwise. 

' This slight comparing of notes as to our own feelings and 
your poetry is not designed to draw on any discussion of dog- 
matic theology. But I remain, my dear Aubrey, affectionately 
yours.' 

From AUBREY DE YERE to SIR W. R. HAMILTON. 

' DINGLE, September 7, 1855. 

* It is a very great pleasure to me to find that you like the 
poems of my new volume which you have had time to read as 
yet; and certainly, considering how short a time you have had 
the book in your hands, you. seem to have made yourself marvel- 
lously well acquainted with its contents, as well as to have read it 
in a most friendly spirit. It is curious that you should have 
chanced to see the poem on Hi yh land Mary. It was written 
during a tour I made in Scotland in the winter of 1840. A 
volume of Burns was the companion of my solitary way. Both 

D 2 



36 Life of Sir William Roivan Hamilton. [1855. 

for him and for the Scotch people, with their singular mixture of 
imagination and warm feeling, with their robustness, courage, and 
perseverance, I felt a great, though by no means unqualified, 
admiration. I wished to write something in honour of the hard 
and of his country ; and it struck me that, by making " Highland 
Mary" the centre of the poem, and, as it were, a connecting link 
running through it, something of heart might find its way into- 
the poem, as well as a certain degree of unity, without which it 
would " drag its slow length along" only to weary the reader. It 
was a sort of farewell tribute to a land in which I had met kind- 
ness, and from which I had received many impulses of thought 
and feeling. A Farewell to Naples, written in a different tone, 
you will find in the same volume. It is probably too severe, but 
ought at least so far to confirm what I say in praise of Rome and 
the other sanctuaries of Italy, as to show that that praise comes 
from one who does not fancy perfection in every city which is 
washed by the Mediterranean, and professes the Roman Catholic 
faith. By-the-way, I am particularly glad to find that you like 
the spirit in which I have touched (at the end of my volume) on 
points respecting which we should differ. My firm conviction 
that the Church to which years of thought made me believe it my 
duty to make my submission (to join a Church is a phrase I could 
never use) represents the authentic and permanent form of Chris- 
tianity nay, that she alone gives to the bodies which have de- 
parted from her whatever of clear, firm, and orthodox teaching 
they retain this belief ought surely to produce no bitterness of 
tone in the expression of it. To leave such convictions ujic.rpressed 
would, on the other hand, imply little regard or respect for others, 
or else very little faith in truth as the great gift to man, through 
which God is pleased to dispense His other gifts. It is, however, 
a mere absurdity to fancy that, by expressing such convictions in 
a rude or bullying tone, we can convey them to others, or that we j 
have a right to be surprised at others still thinking much as we 
once thought. We have neither to impose our sentence as a, judge, 
nor to plead in the spirit of an advocate, but to /r/Vw., in sim- 
plicity and in manly humility, to whatever of truth God has been 
plea*"! to impart t. us, ;in<l thru 1,-f that witness i'niv as it may 
in a world preoccupied and not zealous for truth, though eager I 
enough for 



AKTAT. 50.] Three Feelings. 37 

* To come back, however, to your remarks on my poetry. . . . 
Whenever you have time to go through the volume, you may be 
sure that any remarks you may make on it will be most acceptable 
and useful to me, and that whether or not they should prove as 
favourable as those of your last letter. There are two sets of 
poems on which I should especially like to know your opinion. 
The first is a series of poems on Ireland, printed consecutively, 
and beginning with some sonnets called Colonization. They con- 
clude with a poem called The Last Irish Confiscation, which, being 
cast in a dramatic form, as the address of one of the early and dis- 
possessed race to one of the conquering race, reduced in their turn, 
can be judged on purely poetic grounds, and with or without the 
aid of political sympathy. The note at the end of the volume is 
necessary to explain it. The second class is the series of love 
poems called Psyche. One of them, beginning, " Such beauty was 
not born to die," would, I think, have a special interest for you. 
My poems have been said, even by indulgent readers, to be defi- 
cient in passion. If so, I should think that the defect exists less 
in those two series than elsewhere. I have sent on your remarks 
to my mother and sister, who will be much interested by them.' 

From SIR W. B. HAMILTON to AUBREY DE VERB. 

t OBSERVATORY, September 10, 1855. 

' . . . I have been as happy in my own marriage as I expected, 
and more than I deserved to be. My three loves have been of kinds 
entirely different, and were felt all along to be so. I do not think 
that I ever confounded the three feelings, though it might be 
tedious, and in some degree impertinent, or presumptuous, for me 
to pretend to analyse them now. In general I might perhaps be 
permitted to say, to you, that they sometimes suggest themselves 
to my mind, as having been characteristically those of a lover, a 
brother, and a husband: selecting, you know, what has been emi- 
nent in each of them. 

* I ought to bo packing up and otherwise preparing for Glas- 
gow; but must add one line to thank you for having pointed oat 
Psyche to my notice. I had feared that the poem was a long alle- 
gory, and had resolved to take my time about reading it. The 
thoughts and words are beautiful. 



38 Life of Sir William Rowan Hamilton. [1855. 

' . . . But as for me, I am going rather for amusement and to 
see people than to do anything important at Glasgow. Still I 
would go for the mere purpose of being there on Monday next 
during that single hour when the arrangements respecting the 
place of meeting for next year will be decided on ; in order that, 
as a Dublin man, I may put in once more a claim for a visit of 
the Association to Dublin in 1857.' 

From AUBREY DE VERB to SIR W. E. HAMILTON. 

4 MOUNT TBENCHARD, FOYNES, September 18, 1855. 

[Eef erring to a letter of old date from a friend, expressing 
admiration of Mr. De Vere's character, which had been sent to him 
by Sir W. E. Hamilton.] 

' . . . Tennyson's new volume, Maud, contains these lines 
describing the feelings of a man, when, at his desire, the lady of 
his love, with whom he has quarrelled, gives him back his 
presents : 

" As looks a father on the things 
Of his dead son, I looked on these." 

Such are the feelings with which we are apt to look on memorials 
of early praise and early expectations. There are doubtless, how- 
ever, some who can regard the mementos of their youth with less 
mingled feelings : those persons especially of whom Wordsworth says 
that they "have wrought upon the plan that pleased their childish 
thought," and whose earliest aspirations have brought forth sixty- 
fold or a hundredfold. The chief misfortune of my life I regard as 
having been the absence of the habit of Confession, owing to my 
not having been brought up as a Catholic a habit which I regard 
as necessary in an equal degree, though for different reasons, for 
the most opposite characters the weak and the strong, men of 
genius and the dull. Without it, sterile soils are cheated of the 
increase possible to them, and rich soils abound as much in weeds 
as flowers : our very gifts become special sources of temptation and 
aberration; and we are left to learn, by the experience of litV, 
lemons which ought to have been the foundation of a manly and 
Christian life. Here is a little bit of condensed biography on my part 
in reply, or part reply, to a portion of your letter ; for I speak from 
observation and experience, not theologically, and with reference to 



AETAT. 50.] Three Worlds. 39 

the human aspects of the question only, as a psychological problem, 
without discussing the question whether or not a Rite of Foryirr- 
ness stands to post-baptismal sin, as Baptism stands to Original 
sin. . . . Yet no suffering is sent in vain. All trials find their 
place in that great scheme of Probation which is the Reality of 
Lite in its present stage . . . and all that we have thought, and 
1'clt, and done, lives on in us to the end of our lives, furnishing 
new materials for a probation which is ever changing its character 
in the successive epochs of our life.' 

From SIR W. E. HAMILTON to AUBREY DE VERB. 

* OBSERVATORY, October 19, 1855. 

' 1 found that it at times agitated me to a degree which was 
imprudent for health, of body and of mind, to write as I was 
doing before I went to Glasgow, on subjects that are still so very 
vividly remembered. My visit was an useful diversion of my 
thoughts . . .' 

From AUBREY DE YERE to SIR W. E. HAMILTON. 

'October 20, 1855. 

* ... I hope you will go on writing sonnets, so as to make 
them embody a large proportion of your views and feelings on 
personal, ethical, and philosophic subjects; but I have a still 
more important work* for you which you must execute some 
day, and for which you have such great qualifications . . . Your 
visit to Glasgow must have been a pleasant one and particularly 
useful at the time it took place ; for there is always an agitating 
interest in a review of the critical passages of one's past life, as 
well as a deep moral interest that allays agitation. . . . 

' . . . There is something very touching in what you allude to 
respecting your pilgrimage to the house in which you bad first 
been greeted by the "beautiful Vision." But surely all such 
Visions should be looked on as Anticipations and Types of the 
Glory and Beauty unrevealed, rather than as lights which have 
melted away into the sad shadow-land of the Past. There are 
three Worlds those of Sense, of Faith, and of Glory. The 
first is in some measure eclipsed, or passed by, for a Christian. 

* Not here indicated, but see infra A. De V.'s letter of May 11, 1856. 



4O Life of Sir William Rowan Hamilton. [1855. 

The last is not yet revealed : but all the shapes on Earth that are 
shapes of Beauty and Brightness (flowers, stars, poetry, and what- 
ever belongs to the Ideal) seem to me to be sparkles or flashes shot 
down through the dim atmosphere of the region of Faith from 
the region of Glory. They are gleams to remind us of a higher 
world, just as there are intimations from the region of Faith that 
transpierce the darkness of the lower region of Sense, and give 
a vitality even to paganism. It seems to me of the essence of 
these things that they should be but gleams, for the World that 
unites Brightness and Permanence is yet to come.' 

From the SAME to the SAME. 

' MOTTNT TEENCHAED, December 20, 1855. 

1 . . . I am sending you a copy of a poem which has, over and 
above its poetic merits, an interest for us, Irish readers, being a me- 
morial of a most promising young Irishman, the late Lord Belfast. 
I want you to tell me also how you like it, especially the singular 
effect of the triple rhymes in the proem. The author* has pub- 
lished also a volume of poems, many of which are very beautiful. . . . 

Talking of poetry puts me in mind of your fair friend " who 
Speranza hight." She certainly must be a woman of real poetic 
genius to have written anything so beautiful, and also so full of 
power and grace as the poemt you showed me. For the sake both 
of poetry and old Ireland you must do all you can to make her go 
on writing, and publish a volume soon. Do not forget to tell her 
that you showed her poem to a stranger (a stranger always counts 
for something) who has been addicted to poetry all his life, and re- 
veres it more every year, and that he felt a very sincere admiration 
for it ; although in the matter of your proposed metrical changes, 
he had the impertinence to agree with you.' 

From SIR W. B. HAMILTON to AUBREY DE VERB. 

* November 14, 1855. 

' I do not feel in the least afraid, for myself, of reading any- 
thing. I have read atheistical books, infidel books, Socinian books, 
Protestant evangelical books, books by the Archbishop of Dublin, J 

Denii Florence Mao Carthy. t Shadows from Life. J Archbishop Whatcly. 



AETAT. 50.] The Nichols. 41 

Protestant Uigh-clmrch books, Romanist books; and remain a 
moderate member of (what we used to call) the Church of Eng- 
land : more nearly perhaps approaching to the Archbishop's school 
than when we talked most about such matters formerly, though 
retaining a sincere sympathy and reverence for some whom we 
called Puseyites, and indeed for Dr. Pusey himself, though I am 
more conscious now than before of being farther removed from 
Rome than he is. But I quite distrust my own power of expressing 
to you, with so much of personal delicacy as you have used towards 
me, any of those beliefs of mine which may, or rather must, be 
different from yours. Indeed we should not meet on equal terms: 
for while I could listen to anijthimj from you (though I see no me 
in listening to it), you not merely might be, but (I should almost 
say) ought to be, offended if I were to express at all fully my 
views of the Romish Church. Merely in my calling it such, you 
have already ground for a very pretty quarrel ! though I think 
you will not avail yourself of it this time.' 

I now give extracts from the correspondence with Dr. Nichol 
and his son, commencing with a letter to the son : 

From SIR W. R. HAMILTON to JOHN NICHOL. 

' ON BOARD THE "ARIEL," September 26, 1855. 

* ... It will gratify me much if you shall sometimes send me 
copies or draughts of your poems. I do not pretend to be a poet, 
or in any important sense an original thinker myself, but if you 
knew more of my (very unimportant) private history, you would 
be better able to understand that, without giving up any one 
of my own convictions, I may be able to sympathise with young 
men who earnestly think for themselves, even if the process lead 
them quite up or down to the point of Atheism, to which it does 
not seem to have led you and from which I recoiled. 

* Allow me to repeat, in writing, what I half said in a recent 
conversation, that, much as I love and honour your father, my 
regard for you has a distinct and independent root ; and that 
though I may prove a very bad correspondent to you (in the case 
of Aubrey De Yere there was a whole host of early recollections to 
be called into play, if our remembrance of each other in later times 
had flagged), any letter with which you may at any time favour 
me will interest me for its own sake. 



42 Life of Sir William Rowan Hamilton. [1855. 

' I trust that you cannot be displeased by my adding that if 
Miss Nichol, your sister, should ever favour me with the sight of 
any of her manuscripts, the kindness will be appreciated, and 
in some slight degree repaid by a free but friendly criticism. 

4 With regards to Dr. and Mrs. Nichol,' &c. 

A memorandum, bearing date the same month as the above 
letter, records in the following words Hamilton's modest estimate 
of his poetical faculty : 

* 1855, September. I am just enough of a poet myself to be 
enabled thereby to sympathise with other poets, to enjoy habitually 
their successes, and sometimes to be, without unkiudness, sensible 
to their faults, or short-comings, better than without some such 
exertions of my own (in the department of poetical composition) 
might have been possible for me/ 

From the SAME to the SAME. 

' OBSERVATOKY, DUBLIN, September 29, 1855. 

' My son Archibald (the younger of my two boys) made out, 
this morning, the cipher on the other half of this sheet, without 
his having previously the slightest notion of the subject, or even the 
language ; but I had previously taught him my key. I shall per- 
haps ask him, after a day or two, to write out the same passage 
for you in some other form, consistent with the same key (not 
alphabet), and, like the present, definitely interpretable. You 
would be astonished, I think, and even your father, with all his 
knowledge of the powers of numbers, would be so, if I were to 
mention in how many distinct ways the same extract from your 
poetry could be written on my plan, without the slightest ambiguity 
of interpretation ! A recent and hasty calculation leads to the con- 
clusion that the expression of the number, on the usual decimal 
By stem, would require 230 figures. And even my own initials 
could be written in more than 300 millions of different ways, so 
that my son (for instance), or anyone else who had the key tho 
plan which cm easily be carried in the memory, and is so carried 
by me, would decipher, in a minute (all helps being ready), any 
one of those 300 millions of distinct monograms, as easily as any 
other of them, to signify precisely " W. R. II.", and nothing else 
whatever; and this independently of any arbitrary punctuation, 



AKTAT. 50.] Hamiltorf 3 Cipher. 43 

or introduction of astronomical, or mathematical, or Greek, or 
Grerman, or Oriental symbols, such as might at pleasure be mingled 
with the characters of the English alphabet.' 

From J. P. NICHOL, LL.D., to SIR W. E. HAMILTON. 

* OBSERVATORY [GLASGOW], September 28, 1855. 

* ... My letter to you is, as you will find, little more than a fond 
recalling of the most pleasant week you spent with us. I assure 
you it is a week that none of us will ever forget. You taught 
us far more than your cipher, viz. how exquisite and lovable is 
the highest genius.'. . . 

From SIR W. E. HAMILTON to J. P. NICHOL, LL.D. 

' OBSERVATORY, DUBLIN, October 1, 1855. 

********** 

* The cipher on the other side seems to me likely to be found 
a difficult one to make out, so long as the key is not given, or at 
least the plan explained : an almost unlimited flexibility as to the 
irriting in this cipher, combined with an absolutely rigorous de- 
finiteness in the reading of it, seem to me to be the chief character- 
istics of the method ;....! claim to be able to tea eh the method 
to any sufficiently intelligent person in something like five minutes : 
at least if I should be allowed to use a very simple machine which 
I have invented for the purpose of writing and reading the cipher, 
but with which I can upon occasion dispense. The principle cannot 
be forgotten, and the key can be at any moment reconstructed, or 
the machine re-made, with ease. ... It occurred to me about ten 
years ago, but the war has since made it surge up into my thoughts 
again for it might, perhaps, be useful on some occasion of a secret 
despatch. . . .' 

From SIR W. E. HAMILTON to JOHN NICHOL. 

'OBSERVATORY, DUBLIN, October 10 , 1855. 

* ... When you know me better, you will be aware that I am 
;m abominably bad correspondent, except now and then by fits 
imd starts. To Aubrey de Yere I had not written for several 
years; but, within two or three recent months, he and I have 
exchanged letters, which would make (in size) a respectable little 
volume, though they were (and are) in no way designed for 



44 Life of Sir William Rowan Hamilton. [1855. 

publication, and indeed were in many respects confidential; for 
we wrote much to each other about the days of long ago, being at 
least twenty-five years acquainted, and having, from some circum- 
stances of my history, become intimate almost immediately . . . 
You and he are pretty nearly antipodes of each other, in many 
important respects, yet I can most sincerely sympathise with each 
of you each being earnest and sincere, and also cultivated and 
poetical. That neither is specially scientific may (who knows ?) 
be another attraction; though, that I can honestly enjoy the fame 
of a scientific brother if this last word may without presumption 
be used by me will be seen, if I shall ever transcribe my sonnet to 
the deceased Fourier (author of the mathematical Theory of 
Heat, &c.), or one of those to the living Herschel. Still, the two 
names they mentioned are of men who have been much more than 
mathematical calculators : and your father has such a vast variety 
of cultivation, that one may be pardoned for sometimes forgetting 
that he is by profession (and of course in reality) a man of science. 
Professor De Morgan, of London, is again a person excessively un- 
like to all of you ; yet he and I have exchanged a great number 
of pleasant letters, partly no doubt on mathematics, but we are 
not afraid to write nonsense to each other; at least I send him 
nonsense at times, and he sends me back wit in return, rising 
occasionally to humour. I wonder whether I shall ever venture 
to show you (if I can lay my hand upon it) that letter from Mrs. 
Flamsteed to the late Francis Baily, which De Morgan professed 
to find among the papers of the latter! Flamsteed, you know, was 
the first Astronomer Royal at Greenwich, and a quarrel between 
him and Newton has even recently become matter of discussion. 
Baily died only a few years ago, but he had taken Flanisteed's 
part in the controversy. Mrs. Flamsteed writing (if I remember) 
from 3, Paradise How (or perhaps it was, from 3, Astronomer's 
Row, Paradise), tells Baily that her good man is just gone out to 
take a social stroll with Sir Isaac, with whom he has become quite 
reconciled ; but that, as to that scoundrel, Halley, who had come 
between them, it is lucky for him that he had not come up THI.KI , 
f<>r her husband would have beaten him within an inch of his life ! 
(after all, J I alley, whose name has been given to a comet, was in 
many respects a fine fellow.) I never write in that style myself, 
but certainly I enjoyed De Morgan's report of that celestial- 



AETAT. so.] Estimate of John NicJwl. 45 

infernal letter though I did not show it to my wife, who would 
(or at least might I was not sure) have said it was profane. Still 
she has been much amused with many of De Morgan's letters, and 
I think would welcome him here.* 

* If there be any logic at all in this gossip about my correspon- 
dents, or some of them, it tends to show that I shall be pleased to 
receive at any time a letter from you, whom it is scarcely praise, 
and certainly not flattery, to say that I regard as a remarkable 
young man. Had I only met you in a steam-boat on one of those 
extemporised committees about poetry, I persuade myself that I 
should have long remembered you. As it is, with all the associa- 
tions of your father's fame, and recollections of my reception at his 
very pleasant home, I feel certain that I shall never forget you. 
Yet I am aware that it is TOO LATE for me to hope ever to enjoy 
with you that kind of friendship, which might, nay (I am sure) 
would, have resulted if it had been possible and permitted for us to 
meet as two young men. On that subject, I think that I must 
extract for you a passage from one of my friend Aubrey's letters. 
Meanwhile, you will just write to me as seldom, or as often, as you 
shall feel inclined ; and will remember me with patience and good 
humour, even if I shall leave some interesting letter of yours for 
months, or years, unanswered, though I think that I shan't be 
quite so bad as all that ; but I do take fits on me, at times, of soli- 
tary study, during which I cannot bear to write a letter, and then 
I grow ashamed to write at all. ... 

1 About my cipher (only one of my boys as yet knows it a 
cipher and a mystery seem to be congenial) you really deserve 
credit for your guesses. You are, what in some childish game is 
called " hot" close on the scent ; but have not quite caught the 
hare as yet. You deserve that I should enclose the translation I 
do not say the key. In finding out the latter, I enjoy the thought 
of your having some trouble still !' 

From SIR W. E. HAMILTON to J. P. NICHOL, LL.D. 

' OBSERVATORY, DUBLIN, October 17, 1855. 

* I don't know what puts it into my head, but I am in a mood 
to remember, and write down for you, the latter part of a sonnet 

* See Professor De Morgan's Budget of Paradoxes, p. 188. 



46 Life of Sir William Rowan Hamilton. [1855. 

which I composed by starlight about seven years ago, in the higli 
gallery of the great telescope at Parsonstown.* The former part 
of that sonnet has happily escaped my recollection, but I know 
that it was execrable. Certainly I did not intend to flatter Lord 
Rosse, to whom I had the honour of being introduced by his father 
a great many years before; but there was something of compli- 
ment, or at least of politeness, in my mood, at first, which ought 
not to be, or rather cannot be, the inspiring spirit of any true and 
genuine poetry. Yet I own that I rather like, on recollection, the 
last six lines, expressive of an astronomical enthusiasm, including 
also a feeling for natural beauty, into which my tone gradually 
deepened, and in which I know that you can sympathise. I re- 
member that I was standing alone at midnight, on a sort of little 
bridge somewhat like sixty feet above the ground, while Airy, 
with Lord Rosse, was seeing, at a lower altitude, some newly-dis- 
covered spiral nebula.' 

' Pursuing still his old Homeric march, 
Northward beneath the Pole slow wheeled the Bear : 
Rose over head the great Galactic Arch ; 
Eastward the Pleiads, with their tangled hair : 
Gleamed to the west, far seen, the lake below, 
And through the trees was heard the river's flow.' 



From the SAME to the SAME. 
[FROM A COPY.] 



4 OBSERVATORY, DUBLIN, October 31, 1855. 
'All Hallows' Eve. 



1 ... I really must write again to Agnes [Nichol] to tell her a 
little more about the budding "authorship" of my Helen. It 
takes me quite by surprise, though it may never ripen to anything 
important for the world. She poured upon me, a few days ago, 
quite a stream of questions about your daughter: what sort of 
authorship she was engaged upon, and soforth; but all that may 
better be told to your Agues herself. I assure you that a thought 
of my daughter's about the Sea and its "Despair," which (with 

Vol. ii., p. 620. 



AKTAT. 50.] The Sea's Despair. 47 



glowing cheek pressed to my own) she confessed to me last night, 
appeared to me to be perfectly original ! It was at least entirely 
new to me ; and yet it seemed so deep, but true, that if I had ever 
met with it in reading, I could not have forgotten it.' 

In a later letter to Dr. Nichol, referring to this idea of his 
daughter's, he writes ' I know that there is a Latin motto " Co- 
nantia frangere frango": but still the thought of the sea's despair 
seemed to me new.' 



From J. P. NICHOL, LL.D., to SIR W. E. HAMILTON. 

' OBSERVATORY, GLASGOW, November 6, 1855. 

' Would that I could adequately tell you how much I value 
your confidences, and how gladly I welcome even at this com- 
paratively late date in life the approaches of a new friendship 
to say nothing of such a friendship as yours ! I have outlived 
much, far more than I can estimate in value of any kind ; and it 
is something to find, when one is growing almost weary, and in- 
clined to think oftenest of the future, and a final rest, that new 
companions may still be found to take place on benches that have 
been vacated, and willing to pull with one at the oar. But this 
is only the selfish view of the case. I look with still pro founder 
interest on the events of which you have so kindly told me, because 
I would fain learn what those things are that have torn, tried, and 
educated a mind like yours. Is it not possible that when summer 
returns I may induce you to cross over to us again, and that we 
two shall talk of such things and much else, during a ramble 
through Kannock Moor, or by the margin of vast and soli- 
tary Loch Awe? It is in such places, I always think, that it 
is best for the mind to look into its own depths, and to hear of 
the sorrows and triumphs of another. Do keep this in memory. 
I shall do my best to induce Lady Hamilton to spare you for 
awhile then. 

' 1 never heard of your daughter's rare idea, the " Despair of 
the Sea," but how many images it brings up. Tell her from me 
that she has added a new note to that grand music a music 
which generally gives me a deeper sense of infinitude than even 



48 Life of Sir William Rowan Hamilton. [1855. 

the midnight deeps. Depend on it that the mind which oould 
suggest or find that conception will find many more rare ones. 
Agnes sits opposite me just now, writing a chapter of Hermionc. 
We are alone together in the green room. It is very late, and the 
clocks seem to tick loudly. . . . 

* Mrs. Nichol who has been a guardian angel to Agnes is 
not quite well to-night, and has retired. But you are sure that 
her affection and respect always extend to you. 1 

From SIR W. E. HAMILTON to JOHN NICHOL. 

' OBSEKVATOEY, DUBLIN, Nov. 12, 1855. 

'. . . Any criticism on the " Maud " will interest me much, but 
especially any notes on that poem by one whom I do so strongly 
associate with it as I do yourself. Will you believe that it is only 
within the last few days I have read the poem here ! or indeed 
properly speaking anywhere ; but within those days I have read it 
several times ; and think that it was not indolence which prevented 
me from procuring and reading it sooner, but rather the wish not 
soon to disturb the associations with my having heard you read it 
(or most of it) at your father's Observatory. Perhaps I might 
have reposed on that recollection for some weeks longer, if it had 
not been for the shock received on Monday last (November 5th) 
by the perusal of a clever but uncandid article on Tennyson's 
poetry in general, which appeared in the first number (for Novem- 
ber 3rd) of a new London publication, the Saturday Review, a 
specimen whereof was sent to me. I had recently been reading 
through, and more than once, the " In Memoriam": you conceive, 
from the connexion, that I do not speak here of your lines so en- 
1 it led, though I have lately perused them also with much interest, 
but of Tennyson's, whose Princess, and several shorter poems, I 
had also not long since read again. Consequently, I was the more 
prepared to form my own opinion, as I went along, about the 
justice and candour of the reviewer's criticisms ; and if you wish 
to lash yourself up to some degree of indignation, I recommend 
you to expend sixpence on procuring the article .... One artifice 
of the critic is to treat Tennyson as speaking in his own person 
throughout for instance, in LocMcy Hall, and Maud: as if Shake- 
speare were to be identified with lago ! . . .' 



AETAT. 50.] Despatches from Russia. Logic. 49 



From the SAME to the SAME. 

' OBSERVATORY, DUBLIN, November 24, 1855. 

* ... I am glad that you liked the sonnet to Fourier. Fourier 
was (as your father will tell you) a true poet in Mathematics, and 
in the applications of mathematical science to nature (especially to 
the theory of heat). So was (though not Laplace) Lagrange, to 
whose memory I consider myself as having inscribed those essays 
on a general method in dynamics, for which I received long ago a 
diploma, with an enormous double eagle engraved, and the names 
of Nicholas and of Hamilton stretching about equally across the 
sheet, from the Imperial Academy of Sciences at St. Petersburgh, 
which body has sent me a large packet during the war. There is 
something inspiring and consoling to humanity in the friendly 
and appreciating hand thus stretched through battle across 
Europe ! I astounded a Viceregal party by telling them that 
I had just received despatches from St. Petersburgh, and high 
people spoke, half seriously, of arresting me; but I appealed to 
Lord Carlisle, who was so polite as to take charge of, and after- 
wards return, a specimen of my Russian bulletins. When we next 
met, the Lord Lieutenant asked me whether I had since been en- 
gaged in an animated correspondence with St. Petersburgh ? " No, 
my Lord," I replied, " my last despatches have been from Rome." 
"And your next," said His Excellency, "will be, I suppose, from 
China." I bowed, and had the grace not to answer, that "I should 
not be excessively surprised ! " America noticed me very early. 

' Egotistical all this, is it not ? To be sure it is ; and do you, in 
revenge, make your reply as egotistical as you can find it in your 
heart to do. You will write to a safe and not an inquisitive person, 
who likes to know something of the inner history of any one whom 
he thinks of with regard, but is quite satisfied with the suppression 
of names. . . . Affectionately yours.' 

From the SAME to the SAME. 

1 OBSERVATORY, November 26, 1855. 

* I hoped to write more fully on the logical question, but have 
only been able to give it a corner of my thoughts. The spirit of a 
remark made at the end of a recent note is that we can't go on 
VOL. m. E 



50 Life of Sir William Rowan Hamilton. [1855. 

proving for ever ; we must, it seems, assume some principles ; and 
indeed in each single syllogism the premises of that syllogism ;ii" 
given, not proved. It is not by syllogism, but by insight ami 
sagacity, that the general rules of syllogism itself h&ve been estab- 
lished; though I nm not prepared to deny that any argument 
whatever may be stated in the syllogistic form. That " domino " 
can, I suppose, in the masquerade of thought, be slipped over anv 
one of the intellectual characters which figure at the ball ; whether 
usefully or not is a different question. Whenever we reason, 
are conscious that we generalise ; some principle or tfs-sumption (the 
major) is, or at least MAY be extricated, as we feel sure, from tho 
perhaps individual instance which we have first in view : we regard 
that case (if it be one) as a mere illustration or exemplification, of 
a principle: and then comes in the s?f/>- sumption (the minor), 
whereby we apply what had been formerly generalised, and so 
arrive at a conclusion. But it does not seem to me natural, even for 
a long-trained thinker, to throw arguments into the Aristotelian 
form ; which yet I would advise you to study very diligently, as 
I once did myself. My first science honour in Dublin was won in 
logic. 

* I have just now noticed in that wonderful Almagest of 
Ptolemy the world's astronomical Bible for almost 2000 years 
the word o-iAAoytcr/joc, applied to mathematical calculation. Hobbes, 
as perhaps you are aware, regarded all reasoning as a calculation. 
But I must have forgotten a whole Library ! ' 



AETAT. 50.] Visit to Cheltenham. 5 1 



CHAPTEE XLI. 

VISIT TO CHELTENHAM. ICOSIAN CALCULUS. CORRESPONDENCE 
WITH AUBREY DE VERB. LETTERS ON MATHEMATICAL 
SUBJECTS. 

(1856.) 

DURING the first half of 1856 Lady Hamilton was seriously 
indisposed with illness of a nervous character, similar to that by 
which she was affected in 1840-41, and letters of her husband 
prove not only his natural anxiety on the subject, but his affec- 
tionate devotion to her of personal care and detailed consideration. 
He encourages her pastor, Dr. Sadleir of Castleknock, to enter into 
religious conversation with her, urging with truth and wisdom, * I 
have known even the body benefited by a courageous reference to 
things of a higher life ' : he mentions incidentally, ' I have been 
walking nearly the whole of this morning with Lady Hamilton in 
the garden ' : and he remains in doubt till the very last day, 
whether he shall allow himself the anticipated pleasure of at- 
tending the Meeting of the British Association and being a guest 
of his old friend John Graves at Cheltenham, telling his ex- 
pectant host (August 8) that ' Lady Hamilton has been really very 
ill for a good while past. She has often parted with me before, 
but for the last six months or nearly so, I have been a sort of 
nurse to her, and it is a great effort to her to part with me, at 
present, even for a few days/ This illness of Lady Hamilton, and 
the confinement to which it doomed himself, had undoubtedly an 
injurious effect upon Hamilton's own health and spirits. There 
are indications of this in his journals and letters, prompting in the 
mind of the reader uneasiness and expectation of some break-down, 
Happily he was able at the last moment, on the 10th of August, 

E 2 



52 Life of Sir William Rcnva n Ham ilton. [1856. 

to obtain a release from home duties, and arrive at Cheltenham 
before the close of the Meeting. He was thus enabled to gratify 
his desire publicly to manifest his feeling towards Dr. Lloyd by 
seconding Whewell's proposal that at the succeeding Meeting of 
the Association in Dublin his old friend should have the honour of 
being its President. 

Mr. Grove, the eminent author of The Correlation of Physical 
Forces, was his fellow-guest in the house of his friend ; and familiar 
intercourse with these brothers in science, and more cursory meet- 
ings with other distinguished men, had the happiest effect upon his 
spirits, affording him the enjoyment and stimulus of intellectual 
companionship ; but the physical effects of the disadvantageous 
circumstances I have referred to were not to be escaped from, and 
showed themselves in a severe fit of gout, by which he was ren- 
dered unable to walk, and obliged to remain for a fortnight longer 
than he had intended a recipient of his friend's hospitality. By 
both host and guest this was considered to be the reverse of a 
penalty. Letters given below show how he feasted upon the 
contents of the rich scientific library of Mr. Graves, and, as in 
earlier times, received from his friend intelligent sympathy in 
his most recent researches, and hints for prosecuting them in 
new directions. 

Very soon after his return home he forwarded to Mr. Graves 
two valuable scientific Papers : the first a criticism on Schefner's 
book, Der Situations Kalkul (a work on the Geometry of Space), in 
which the author arrives, after long discussion, at a conclusion 
which in 1832 had been rejected by Hamilton as insufficient founda- 
tion for the required calculus ; the second, an extension by means i 
of quaternions of some propositions laid down by Gauss in hisj 
Disqumtioncs Arithmeticae. Their general nature is set forth in| 
letters to Mr. Graves, which are, however, too mathematical fc 
insertion in this volume: the papers themselves will probablj 
appear in an early number of Hermathena. 

In the following month of October the happy birth-month 
both Conical Refraction and Quaternions Hamilton imparts 

I 



AKTAT. ,V).] The Icosian Calculus. 53 

the same friend his discovery of a new Calculus, to which he gave 
the name of Eicosian (afterwards changed to Icosian), because 
specially exemplified in the case of the Icosahedron, though 
applicable to other polyhedra. This offspring of his genius was 
not indeed of equal note with those which have been named, but it 
is not unworthy of its parentage, j Professor Tait says of it 
* Hamilton has published but a page or two with reference to 
them [the systems embraced by this discovery], yet that little is 
enough to show the probability of their becoming, at some future 
time, of great importance in the study of crystals and polyhedra 
in general/ fit was Mr. Graves who, during Hamilton's visit to 
Cheltenham, liad re-excited in him an interest, long dormant, 
in the mathematical properties of these solids, and to Mr. Graves 
accordingly Hamilton first made known his discovery, in letters 
dated October 7 and October 17, (of which the latter extended to 
fifty pages. A summary view' of it was communicated by him, 
under date October 29, to the Philosophical Magazine for December, 
1856, with the title Memorandum respecting a New System of Roots of 
Unity, and a fuller statement of the discovery was made by him to 
the Royal Irish Academy on the 10th of November, and is re- 
ported in the Proceedings. From these communications I extract 
the following passages, which will give the reader a general idea 
of the Calculus : 

* I have lately been led to the conception of a new system, or 
rather family of systems, of non-commutative roots of unity, which 
are entirely distinct from the ", /, k, of the quaternions, though 
having some general analogy thereto ; and whioli admit, even more 
easily than the quaternion symbols do, of geometrical interpretation. 
In the system which seems at present to be the most interesting 
one [that dealing with the icosahedron and the dodecahedron] 
among those included in this new family, I assume three symbols, 
i y K, A, such that 

c; (A) 



where IK must be distinguished from KI, since otherwise we should 
have A fi = 1, A = 1. As a very simple specimen of the symbolical 



54 Life of Sir William Rowan Hamilton. [1856. 

conclusions deduced from these fundamental assumptions, I may 
mention that if we make 

fJL = IK* = A(A, 

we shall have also 

^' = 1, A = ^; 

so that IJL is a new fifth root of unity, connected with the former 
fifth root A by relations of perfect reciprocity. A long train of 
such symbolical deductions is found to follow ; and every one of 
the results may be interpreted as having reference to the passage 
iromface to face (or from corner to corner) of the icosahedron (or of 
the dodecahedron) ; on which account I am at present disposed to 
give the name of the " Icosian Calculus " to this new system of 
symbols, and of rules for their operation.' Philosophical Magazine. 
' This Calculus agrees with that of the Quaternions in three 
important respects namely, 1st, that its chief symbols, t, ic, A, are 
roots of unity, as t,/, k, are certain fourth roots thereof; 2nd, that 
these new roots obey the associative law of multiplication ; and, 
3rd, that they are not subject to the commutative law, or that their 
places as factors must not in general be altered in a product. ' And 
it differs from the Quaternion Calculus 1st, by involving roots 
with different exponents ; and, 2ndly, by not requiring (so 
far as yet appears) the distributive property of multiplication. 
In fact + and -, in these new calculations, enter only as connect- 
ing exponents, and not as connecting terms : indeed, no terms, 
or, in other words, no polynomes, nor even binomes, have hitherto 
presented themselves in these late researches of the author. As 
regards the exponents of the new roots, it may be mentioned that in 
the principal system for the new Calculus involves a family of 
systems there are adopted the equations, 

1 = ,* = K' = A 5 , A = < K ; (A) 

so that we deal in it with a new square root, cube root, and fifth 
root, of positive unity ; the latter root being the product of t ho 
two former, when taken in an order assigned, but not in the oppo- 
site order. From these simple assumptions (A), a loog train of 
consistent calculations opens itself out, for every result of which 
1h ic is found a corresponding geometrical interpretation, in the 
theory of two of the celebrated solids of antiquity, alluded to with 
interest by Plato in the Timaut namely, the loosalicdron and 



AKTAT. 50.] The Ic()\ia)i Game. 55 

the Dodecahedron ; whereof the angles may now be unequal. 13y 
making A 4 = 1 the author obtains other symbolical results, which 
are interpreted by the Octahedron and the Hexahedron. Tho 
Pyramid is, in this theory, almost too simple to be interesting, 
but it is dealt with by the assumption A 3 = 1, the other equations 
(A) being untouched. . . .' Proceeding* of li. I. A. 

The investigation by this calculus of the properties of the 
Icosahedron led Hamilton to devise a mathematical game, which 
he called the Icosian Game. A diagram was formed by the pro- 
jection of this exemplar solid upon a flat surface : and the passing 
continuously by the connecting lines from one corner in this dia- 
gram to another became in the hands of the designer a suggester 
of possible sequences, cyclical and non-cyclical, which, amounting 
in number to 62,400 (33,600 being cyclical), were included in 520 
formula), all by him set forth in tabular form. The game founded 
upon the basis of this idea was to be played upon a board, on 
which the figure was cut in grooved lines with holes at the corners. 
In these holes, marked by the twenty consonants of the alphabet, 
were to be inserted consecutively, according to named conditions, 
movable pieces numbered from 1 to 20. The practical objection 
to this game was that, while some of the problems which might be 
set to the player required for their solution considerable thought 
and ingenuity, the majority were too easy to be interesting.* He 
corresponded at great length on the subject with Mr. Graves, by 
whom he was put in the way of selling the copyright of the game 
to Mr. Jaques of Piccadilly, who paid for it the sum of 25, the 
only pecuniary reward ever accruing to Hamilton directly from 
any discovery or publication of his. It is to be feared that 
Mr. Jaques did not find the bargain to be a profitable one. 

The calculus itself, and the problems of the Icosian Game, 



* In a note to Dr. Luby, F.T.C.D., dated July 6, 1860, Hamilton writes 
4 1 cannot resist the temptation of telling you that you and Dr. Hart did not 
solve my little problem of the Icosian Board yesterday,' and adds, that having 
had the intention of presenting a copy of his Board to the Common Room of the 
College, he was discouraged by its reputation of being too easy. 



56 Life of Sir William Rowan Hamilton. [1856. 

afforded scope to their author for an amount of labour well-nigh 
exhaustive in its extent, which fills many thick manuscript-books, 
and is calculated to excite a feeling of wonder in all who are 
capable of estimating it. The letters to Mr. Graves forming the 
original communication ought, perhaps, to see the light, particularly 
as the writer was prevented by the preparation of his Element* of 
Quaternions from fulfilling his expressed intention of sending to 
the Philosophical Magazine further communications on the subject. 
Professor Tait concludes his notice of the Icosian Calculus 
and the Icosian Game by the comment ' the analysis employed, 
though very simple, is more startingly novel than even that of the 
quaternions. The i 9 j, k, of quaternions can, as we have seen, be 
represented by three definite unit-lines at right angles to each 
other. How can we represent geometrically the A or the /i of this 
new calculus, either of which produces precisely the same effect, 
whatever edge of whatever face of the dodecahedron it be applied 
to?'* 

By what has been now sketched of the incidents of the year, 
the reader will have been prepared for the correspondence refer- 
ring to them ; but I give their proper precedence to letters which 
passed in January and February between Hamilton and Mr. 
DeVere. The interest of these letters is literary and personal, 
and the frankness, the open-heartedness of both writers, will, I 
think, be felt by every reader to afford the best proofs of the 
mutual trust, the reality and depth, of their very beautiful friend- 
ship. The reader will be best pleased by taking note of these 
proofs himself : I abstain, therefore, from forestalling his pleasure 
by anticipatory indication or comment. 

From SIR W. B. HAMILTON to AUBREY DB VERB. 

' OBSERVATORY, January 1, 1856. 

' . . . I have read the Essays [Essays, mostly Theological, by 
A. I >. V.] quite through more than once, and many parts of tlu-m 



tith Review, Bqptemte, L886. 



AETAT. 50.] Speranza. Sir Aubrey De Vere. 57 

with admiration ; but the chief feeling which they excite in me, 
or the prevailing impression that they leave, is the old " Talis 
cum sis, utinam noster esses," which, perhaps, you at times recip- 
rocate. Your present views seem to suit you at least, and you 
write a capital style. But this must sound to you irrelevant, 
perhaps irreverent. 

* Many thanks for returning my manuscripts. Your last con- 
versation almost tempted me into writing again on the potentialities 
of the past an idle, if not a dangerous theme yet one to which 
the mind will recur.' 



From AUBREY DE VERE to SIR W. R. HAMILTON. 

* CURRAGH CHASE, January 4, 1856. 

1 ... I send you back Speranza's most amiable letter. It is 
indeed pleasant to meet that rare thing, poetic genius, in union 
with a rarer one the magnanimity (in which genius is so often 
deficient, and without which it almost ceases to be respectable) 
which can take censure with gratitude, praise with simplicity, and 
both with equal grace. I am much pleased at the terms in which 
she speaks of my father's poetry, because I think them deserved. 
Wordsworth, a critic hard to please, thought his sonnets among 
the best in the language, both as to matter and composition. 
Notwithstanding, he remains almost as unknown in this country, 
to which he was deeply and loyally attached (in spite of his love 
of England, and dislike of popular agitation and such patriotism 
as is only Pat-^/odsm), as if Ireland boasted poets as numerous as 
blackberries.' 

From SIR W. E. HAMILTON to AUBREY DE VERB. 

* OBSERVATORY, January 15, 1856. 

* I am in your debt at least two letters, and have at this mo- 
ment to thank you for sending to me the newspaper containing 
a sketch of your very interesting lecture on General Literature, 
which, on the whole, appears to have been ably reported, though 
slight typographical faults occur : for instance . . . These things, 
you know, will force themselves upon a practised eye. Your own 
published volumes (and / may be excused for reckoning that you 



58 Life of Sir Will ia i Rowan Hamilton. [1856. 

have laid more than a grain or two upon the hills of gold) are 
singularly free from such faults ; and I too have, or think myself 
to have, contributed a smaller proportion of typographical errors 
to my work on quaternions than is confessed by any equally 
conscientious mathematical author: for example, by Professor 
De Morgan. 

* To turn for a moment to something higher. You know that 
I am not a flatterer ; but I must say that I regard your character 
as nobler than my own. I might easily expand upon this topic, 
which the slight praise given to you publicly of late has so little 
suggested, that it seems to me painfully inadequate. But with 
the same sincerity I must say, that the modern Roman system 
appears to me so corrupt that the inevitable tendency of any other 
system, such as Puseyism, to it is, with me, a " reductio ad 
absurduni." 

* I do not admit that I am a " touchy " person ; the pain, and 
often the meanness, of such a character I count myself to have, 
through life, escaped. Yet I am glad that you have not tried my 
temper, by offering to send me stamps for the prepayment of your 
volume of collected Essays through the post.' 

From AUBREY DE VERB to SIR "W. E. HAMILTON. 

4 CUHRAGH CHASE, January 24, 1856. 

* The volume of my Essays reached me quite safely ; and I 
can assure you that, notwithstanding my joke about the stamps, 
nothing was ever further from my intention than to send them to 
you. There are, however, persons to whom I would have sent 
them regular " nineteenth century " persons, as I should call 
them. When I was in Greece, I chanced to be acquainted with 
two people of good position at Athens one an American clergy- 
man, the other a Scotch lawyer. They were brothers-in-law. 
The Scotchman, of course, was "canny" in his way, but nnu-li 
more for others than himself; and while devoting all his eneiv 

to the cause of Greece (he had fought for that country, as well aa 
laboured for the education of its youth at a later period), remained 
always as poor, not only as a field-mouse, but as a field-mouse on 
u Scotch heath. The American was also an excellent man, wi-11- 
lo in the world, with a fine house, comely wife, high posit inn 



AETAT. 50.] Character of the Scotchman. 59 

in a little world of fashion, large reputation, &c. &c. "Well, 
fancy this man confiding to me one day that his brother-in-law 
wjis not quite the right tiling ! " Would you believe," said he, 
" it is now two months since I had to pay a penny for him one 
day, as we were out riding together, and Jtc //c/.s ncn-r rrj^tid it 
yet!" 

' "When I was in Scotland, I perceived at once, even in the 
Lowlands, that Scotch worldliness is not like American or Eng- 
lish. A Scotchman has warm and deep feelings ; and if he is 
selfish, it is, in part, because out of the abundance of his love- 
power he has a great deal to spend on himself as well as on his 
friends. There are those who care nothing for themselves, and 
care as little for their neighbours. Moreover, the Scotchman, 
however thrifty he may be, has large sympathies, and a strong 
imagination too. He has reverence for antiquity ; loves old kings 
and chiefs even more than modern lords and ladies ; preserves old 
legends ; imagines a something venerable even in that shrill- voiced 
demagogue, John Knox ; sings the songs of Burns on moor and 
heather ; really loves his country, and remembers its ancient fame 
when he looks on Edinburgh Castle, if not, alas ! when he looks 
on Melrose Abbey ; is capable of being profoundly attached to a 
cause, as well as to a person, and in defence of either bears the 
beating of the worldly storm, as the northern side of his native 
pines bears the sleet-blast and the clinging ice. In short, in the 
midst of his coarser nature there is a finer one also, the grain 
of which is delicate as that of those woods of which musical in- 
struments are made. In many of these gentler qualities he is 
excelled by the Irishman : but then he has what we lack a 
robustness like that of the canvas used for storm-sails. 

* I should like to know why I am writing you a disquisition 
on Scotchmen. I suppose it is because I intended to say a few 
words on theology, in answer to your remark on a tendency to 
modern Romanism being a " reductio ad absurdum," just as once 
before I wrote a theological letter to you when I meant to write 
one about literature and poetry. 

'Friday, January 25. P.S. My letter was late for the post. 
. . . My vindication of Scotland was, I suppose, a vindication of 
her from an imaginary charge that of her being implicated in 
the ordinary " nineteenth century " character, with which she is 



60 Life of Sir William Rowan Hamilton. [isse. 

commonly, but most unjustly associated, even more than England. 
I might have remembered, however, that I had done what I could 
in this way in writing my Highland Mary, to which you had done 
so much more than justice. In a large part of my poetry, indeed, 
my purpose has ever been to vindicate some just cause, or to write 
the epitaph of one unjustly trampled down, and scrape the moss 
from neglected tombstones. 

* Now one word as to the "reductio ad absurdum " implied in 
a tendency to modern Romanism, i.e. Romanism seen through 
the mist of modern associations, party impressions, social antago- 
nisms, &c. I will only say that nothing of this kind can be taken 
by a philosophic mind like yours for more than, at most, an im- 
pression as distinguished from opinion, and must even be contrasted 
with conviction.' 

Mr. DeVere then tills fourteen closely- written pages with a 
very ably-argued contention, that the charges brought against the 
Roman Church were necessarily incurred by a spiritual body, 
composed of sinful human beings, acting through many successive 
centuries upon a sinful world, and that they were of the same 
kind as the member of the Church of England felt to be unjust 
when brought against his Church by deserters. He concludes his 
letter by turning from the general argument to the following 
personal application of it : 

* I do not want to urge a new set of d priori impressions on 
you, in place of those so common among us, and to which I was 
once myself servile indeed, though I suspected it not. That would 
not be fair. I only want to point out the absolute necessity of 
shaking off all such vague impressions, and all that is not really 
and truly theological and philosophical in our method of thought, 
if we would reach truth on the most important of all points, and 
discriminate between the authentic and the spurious forms of God's 
Revelation. I have observed that those sorts of impression* and 
social traditions have often even more influence on the thought 1'ul 
than the thoughtless, if they be not on their guard against tlu-m, 
or if they fight the battle of theology with the left hand, whito 
more earnestly interested in political or philosophic avocations. 
Onoe more, adieu. Yours affectionately/ 



AETAT. so.] Non-Version. Henry Taylor. 61 

It could not reasonably be expected that Hamilton, with such 
work on his hands as the writing of his Elements of Quafernfoni, 

and with his thoughts absorbed in it, could find time or freedom 
of mind sufficient to enable him to grapple with so accomplished 
and earnest a controversialist as his friend. In his reply, accord- 
ingly, he is content with intimating that his previous convictions 
remained unshaken, while candidly admitting the fact that he was 
conscious of having his share of bias and prepossession. 

From SIR W. E. HAMILTON to AUBREY DE VERB. 

* OBSERVATORY, January, 31, 1856. 

* ... I like what you say about Scotland, and am not offended 
by your remarks on Theology though not at all afraid of your 
succeeding in con- or per-verting me. In my Lectures on Quater- 
nions there occur all sorts of " Versions " ; such as Aversions, 
P/'oversions, and T?wisversions, with reference to rotations in 
geometry ; but one of them is a Non- version ! Even mathematics, 
you see, may supply a joke sometimes such as Pope somewhere 
says, does he not? that " gentle dulness loves." 

* Of course I admit that early impressions must in my case, as 
in most others, leave a very lasting trace and that I do not pre- 
tend to be unbiassed as indeed who can be ? . . .' 

In rejoinder to the above, Aubrey De Yere discharges a few 
Parthian shots, to be met with towards the conclusion of the 
following letter, the preceding part of which I insert on account 
of its interesting reference to his own poetry, and his, as it appears 
to me, admirable characterisation of the poetry of Sir Henry 
Taylor. 

From AUBREY DE YERE to SIR "W. E. HAMILTON. 

'CuRRAGH CHASE, February 1, 1856. 

'. . . There is no quality I should be better pleased to see 
attributed to my poetry than Passion partly perhaps because 
some not unfriendly critics have thought that its chief defect was 



62 Life of Sir William Rowan Hamilton. [1856. 

a lack of Pass ton. Henry Taylor suggests this in the form of a query, 
in a letter which assuredly is complimentary enough, and which, 
if I can find it, I will send to you (but you must let me have it 
again), though you are not to suppose I believe in all the kind 
things he says also of my verse. I was so ungracious as to answer 
his question by saying that I doubted very much whether it was in 
Passion that my poetry was deficient: but an author is the 
worst judge of his own works. . . . 

* As for the Versions and Non- versions, I will give you, as a 
pious task, the Conversion of our friend Anster on the subject of 
the Poet I have named to you, Henry Taylor. He is flinty- 
hearted as to him, which always amazes me : for Philip ran 
Artevelde seems to me the greatest dramatic work since Shake- 
speare ; and his other poems are admirable also. The degree of 
compact strength they exhibit is almost a miracle in these soft 
effeminate days, when few could claim the praise old Ben Jonson 
gave to a friend of his of whom he said, "my son Cartwright 
writes all like a man." To this H. Taylor adds a peculiar delicacy 
and fineness, of double value when united with strength, and a 
deep vein of genuine Thought, and a certain " noble and stately 
grace " which I should call the differentia of his muse. . . . Anster 
needs conversion also very much respecting Tennyson, respecting 
whom he is but indifferently affected. 

* I must not serve you up any more Theology at present. I will 
only beg of you to remember one thing, viz., that however natural 
it is for us to be biassed by early training, or the "Public Opinion" 
about us, respecting Religion, and also to act on Itnpremm'ons as 
though they were Convictions deliberately arrived at, yet, if there 
be any one thing absolutely certain, it is this, viz., that Duty, 
Honour, and Safety all alike imperatively require that we should, 
as far as we are able, rise superior to this natural tendency, which, 
even were it excusable in any, would not be so in persons of deeply 
thoughtful habits. All this belongs to the Will, for which we 
are plainly responsible ; and if we believe in Christianity at all, 
surely it must be our first duty to discriminate with perfect//// 
between the authentic and spurious versions of it. After all, suivly 
consistent and logical thinking is but honest thinking on such a 
subject; and to many may not this be their chief, if not their only, 
probation ? Affectionately yours.' 



. so.] Dr. T. 7?. Robinson. Romanism* 63 

From SIR W. E. HAMILTON to AUBREY DE YUM:. 

1 OBSERVATORY, February 12, 18.j). 

* I have thought myself permitted to make a copy, in one of 
my manuscript-books, of Henry Taylor's remarks upon your 
poetry, all of which I think that I can adopt. Indeed most of 
them, or at all events many, had occurred to myself before. Per- 
haps I may write to you again upon the subject. At this moment 
I am detaining the morning postman, who is not bound to wait for 
letters, but whom I have bribed by promising him a cup of tea in 
my hall. He has brought me a note from Dr. Eobinson, enclosing 
four stamps in repayment of a recent advance of mine, so if you 
hud sent me stamps for posting your book (which I am very glad 
that you did not), you would not have been the only Irishman who 
has done something of this kind. I am rather vexed by receiving 
these stamps, but suppose that the civil thing is to retain them. 
I had put the same number of heads on a printer's packet which 
came to me by mistake a day or two ago. You will return me 
Eobinson's note. He is a very pleasant person, to love or to 
quarrel with.* I have known him since I was a child, and we 
have had our little quarrels now and then. You and I never got 
up one ! Affectionately yours.' 

Although Hamilton declined to enter into written controversy 
with Aubrey De Yere upon the subject of Komanism, it would 
appear, from an entry in one of his manuscript-books, that his 
friend's arguments had led him to formulate, under the name of 
theses, some of his convictions in reference to the Church of Rome ; 
and I think it right here to print them, for the purpose of showing 
that his resolute adherence to Protestantism was founded not 



* What Dr. Robinson at this time thought of Hamilton is shown by a note 
written by him about a month later in reference to something said by Hamilton 
at a Meeting of Council of the Royal Irish Academy. 

1 OBSERVATORY, ARMAGH, March 8, 1856. There was not the least occasion 
for you to suppose I could be hurt by what you said. I know your kind and 
good heart too well, and rely too firmly on our old friendship, ever to admit the 
possibility of your harbouring an unkind thought towards me ; and 1 took [it] 
as one of t\ie points to which an extempore speaker is occasionally tempted.' 



64 Life of Sir William Rowan Hamilton. [1856. 

solely upon traditional views and prejudices, but upon historical 
study and reasoned conclusions. At p. 349 of Manuscript-Book 
D. 1855, 1 find the following :- 

* On O evening, July 6, '56, I wrote out for myself, and 
showed to Helen, with some dim purpose of hereafter letting 
A. De Yere see them, a few remarks in the form of Theses on 
Romanism. They were nearly these : 

4 1. The Roman Pontiff is not the Vicar of Christ ; nor is there 
any human Head of Christendom. 

* II. The Eoman Communion is not the Holy Catholic Church ; 
nor is it even an incorrupt branch thereof. 

* III. The Eoman System is unscriptural ; and is expressly 
denounced by anticipation in the Scriptures. 

* IV. The Roman Worship is idolatrous in tendency ; and its 
followers are in peril of the sin of actual idolatry.' 

The absorbing nature of his mathematical researches is graphi- 
cally expressed in the following extract from a letter to Aubrey 
De Vere. It renders account for his not giving an unconditional 
promise to look over a scientific Paper written by a gentleman 
introduced to him by Mr. De Vere ; but it will show also how 
entirely his pursuits and his duties disabled him for discussing 
adequately by letter so great a controversy as that between Pro- 
testantism and Eomanism. 

'July 18, 1856 ... In fact it is impossible for me to give any 
promise of that kind, because when a whirlwind of original investi- 
gation seizes me, it carries me away, and for the time my place 
knows me no more I am out of space, and more than " prope 
mancus ad official ' And then I have all sorts of duties, already 
imperfectly discharged, and which I must not unnecessarily com- 
plicate.' 

The following passage in a letter of Aubrey De Vere's, dated 
May 11, 1856, brings into view a project, entertained long before 
by Hamilton, of extending Bishop Butler's analogical argument 
to a defence of the Church, and also a striking similitude conceived 



AETAT. so.] Projected Essay. Suggestions of DeVere. 65 

by him, at a still earlier time, in which a phenomenon of nature is 
made to illustrate a Christian doctrine : 

* ... I well remember the discovery of Conical Refraction, 
though, alas, little able to appreciate these high things in science, 
much as I venerate them. That discovery, however, particularly 
impressed me at the time, as such predictions in science, like pro- 
phecies in religion, have a special value of their own. I wish some 
competent person would write an essay on all the analogies or 
" similitudes in dissimilitude " between the Sciences you are ad- 
dicted to especially and Theology. I remember Manning (a deeply 
philosophical Mind) observing to me soon after he had become a 
Roman Catholic (1) that it was not, in his judgment, possible for 
Christianity to have expanded itself into any large and consistent 
scheme of thought, such as is implied in the idea of revealed truth, 
except in the form of Catholic Theology ; and (2) that next to this 
wonderful fabric of supernatural science (as set forth, for example, 
in St. Thomas Aquinas], the thing that he most venerated was the 
marvellous pile of Inductive and Mathematical Science, which was 
a development not less faithful of their own Laws of Thought. He 
spoke as if the two things, considered in relation to intellect alone, 
and apart from the soul, might be regarded as the twin summits 
of that mount 

" which doth divide 
Into two ample horns his forehead wide." 

Now, there is a subject for you to write an essay on, by way of 
following up a scheme of thought which had presented itself to 
you before the days of Conical Refraction, when you spoke of 
writing an Essay pointing out the complete relevancy of Bishop 
Butler's mode of reasoning to the question of the Church, as well 
as to that of the Gospel. . . . 

* Here is another plan for you if you like poetry better than 
prose at least. I often wish you would write a series of philoso- 
phical sonnets. The condensed form of the sonnet makes it par- 
ticularly valuable, as a sort of chalice for some one great and 
fruitful thought which deserves to be thus brought out. Again, 
at the various periods of human life there is a peculiar interest, 

VOL. III. F 



66 Life of Sir William Rowan Hamilton. [1856. 

personal, as well as philosophical, in looking back with a view of 
ascertaining what are the most precious spoils which a certain 
quantum of meditation and experience has left with us, and which 
we should like to bequeath. I will mention a thought worthy of 
a sonnet, which will show you that I remember past times as well 
as you. Many years ago, when we were walking on the pier at 
Kingstown, you remarked on the contrast between the vast pacific 
expanse of a bright and level sea with the soft thunder of the 
breakers, which fell with perfect regularity, but drawing down a 
weight of ruin which must have destroyed any boat which jtnt 
itself in the icay of such destruction. You remarked on this con- 
trast as illustrating at once the infinite mercy and infinite justice 
of God, and showing how Wrath and Vengeance are consistent 
with a Being all Love, and above all Passion. Now, in these days, 
when people, who mistake themselves for deep thinkers, as well as 
philanthropists, fancy that they can recommend Christianity by 
expurgating it of its doctrine of " Eternal Punishment," and 
render it more acceptable by divesting it of what, when superfi- 
cially regarded, becomes part of the " scandal of the cross," you 
would do real good by treating the subject in a higher, though 
less popular spirit, and the sonnet form would answer well. 
Perhaps several sonnets might be necessary. . . . 

* I have no doubt you are quite right in thinking that no cor- 
respondence is desirable. The more tranquil we can keep our 
spirit, the better will that still lake reflect the image of heaven, 
and the less will be stirred up of whatever can dim the waters. 
Besides, the past is a delicate thing, perfect in itself, but easily 
spoilt.' 

The letters which follow relate principally to Quaternions and 
to Polyhedra, but to them I prefix a letter to Dr. Robinson, of an 
earlier date. 

From SIR W. R. HAMILTON to the REV. T. E. ROBINSON, D.D. 

1 OBSERVATORY, March 24, 1856. 

4 MY DEAR DR. ROBINSON Within the last few days I huvo 
received from Paris a quarto of about 200 pages, almost enthvlj 



AKTAT. 51.] M. Houel. A Paradise of Books. 67 

devoted to the development and application of ray results in phy- 
sical astronomy the first part relating to my abstract results in 
dynamics, and the second being* headed, Thhe (FaQtronomie. Ap- 
plication d<> In 31<'tlio<t<> <(<> M. Hamilton an Cal<'ti/ <lcx PtThirlntinna 
d<> Jn inter by Saturn, Uranus, Neptune and Mars, the Earth, &c. 
witli long inequalities of all sorts extended to the years of 
our Lord, 2300, and 2800 all by Prof, or Monsieur Houel, of 
Alenon but submitted to Cauchy, Duhamel, and Delaunay, 
and (as it seems) approved by them. How comfortable to see my 
abstract results translated into hundredths of seconds sexagesimal ! 
and how odd a feeling it gives to read, in the astronomical depart- 
ment, every now and then, of " 1'ellipse de M. Hamilton " ! or 
better still, here and there, without the " M," " 1'ellipse de Ha- 
milton " ! for it is the truth, though perhaps scarcely two or three 
persons in these countries have noticed it, that I assigned, twenty 
years ago, elliptic orbits for all the planets, essentially distinct in 
theory, though very little differing in practice, from those so beau- 
tifully imagined by Lagrange, and having certain centrobasic and 
symmetric advantages.* ... I remain, in haste, but wishing to 
consult you on some things astronomical, most faithfully yours/ 

From SIR W. E. HAMILTON to EEV. HUMPHREY LLOYD, D.D. 

' OBSERVATORY, DUNSINZ, September 9, 1856. 

1 ... I returned to this place only a week ago, having lingered 
for a quiet fortnight in Cheltenham, after the business and amuse- 
ment of the Association week ; which with me extended from 
Monday the llth, to Saturday the 16th of August, for I was 
asked to several dinners, and visited, after the formal meetings 
had closed, the College, and other show-places, such as Lord 
Northwick'sf splendid picture gallery, during that time. Con- 
ceive me shut up and revelling for a fortnight in John Graves's 
Paradise of Books ! of which he has really an astonishingly 



* See vol. ii., pp. 112-117. 

t I happened to be introduced to his Lordship, who is now old, and he told 
me that he well remembered the English Sir William Hamilton at Naples, and 
showed me a miniature of Lord Nelson and a gem of his Lady Hamilton. 

F2 



68 Life of Sir William Rowan Hamilton . [1856. 

extensive collection, especially in the curious and mathematical 
kinds.* Such new works from the Continent as he has pick. d 
up ! and such rare old ones too ! Besides works of Archimedes 
and Apollonius, which I had read before, he showed me the 
original edition of Copernicus's book De lievolutionibus, &c., con- 
taining an apologetic preface by the editor, which Graves had not 
observed, and which, while modestly putting forward the motion 
of the earth as an kypoihexiti endeavours to deprecate by anticipa- 
tion the displeasure, not of the priests, but of the philosophers ! 
for no fear seems to have been as yet entertained of awakening 
the wrath of the Church ; and indeed I believe that the work was 
dedicated to the Pope of the time, but am not quite sure of this. 

' To descend to more recent times though on my way to them 
I lingered for a good while on a charming folio of the works of 
Wallis, written in part in English first, but afterwards translated 
into Latin, for the greater ease of the reader, and including a 
defence of the Sunday against the Saturday, which latter day has 
(I believe) still some advocates in Christendom, as being the 
Sabbath of the Bible. I was induced to read some modern 
German publications chiefly on the Theory of Numbers, which 
is a favourite study of John Graves, though I have very little 
attended to it. I scarcely knew before I was with him lately 
that theorems respecting real integers have been extended to 
imaginary integers, such as 3 + TV/- 1 under the name of complex 
numbers; and that in this extended view the number 2 (for 
instance) ceases to be prime because it is = the product (1 + */ - 1) 
(1 <y - 1) ; though 3 remains a prime number. Graves pointed 
out to me that, in a future theory of integer quaternions no re 



* The following is an extract from the Preface of the Catalogue of Books 
the General Library and in the South Library of University College, 
1H79 : 'The Graves Library is a most valuable collection of more than 
thousand books, and about half as many pamphlets. Mr. J. T. Graves, M.A., 
who was Professor of Jurisprudence in the College between the years 1 s <^ 
1843, by his will, dated March 26, 1870, devised "all his Mathematical, 
tronomical, and Physical books and papers, to University College, London, 
remembrance of his former connexion with that institution." Perhaps 
private scholar has ever formed a mathematical library so nearly conipK-t 
Many of the books are very rare some probably unique aiid about one-half* 
the whole collection is in handsome bindings.' 



A i- TAT. 51.] Gauss. Scheffler. Grassmann. Moebins. 69 

integer will continue to be prime ; because in quaternions 

ic 1 + x* + y* + z' = (w + ix + jy + kz) x (w - ix - jy kz), 



and criT)/ real and positive integer is known to be the sum of four 
square numbers (0 included). I delighted* him by dashing offf 
a solution of a problem suggested by Scheffler's investigation re- 
specting the "greatest common measure of two complex numbers," 
whicli he had supposed would be found difficult, namely, " to find 
the greatest common measure of two proposed quaternions." He 
named at random, 1 + 2i + 3j + 4k, and 5 + 6j + 7/ + 8A-; and I 
soon assigned (by a general process) i - k as their greatest common 
measure : multiplied (it is understood) by 1, or , or +/, or k. 

' John Graves has given me what it was a wonderful kindness 
and favour for a boob-collector to bestow, a duplicate copy of the 
now rare work of Gauss, Disquisit tones Arithmeticae, 1801, which 
I had the astonishing honesty to return (after keeping it only a 
single year) to Sir John Herschel about ten years ago. It is (I 
understand) very difficult now to procure it. As some slight 
return to Graves, I have written to him a long mathematical letter 
to-day, showing how a few of Gauss's theorems may be extended 
to quaternions with suitable but instructive modifications. 

* Another German book, besides Scheffler's Unbestimmte Ana- 
lytik, Hanover, 1854, which treats of problems respecting real and 
complex numbers founded very much on the great work of Gauss, 
and containing the solution of the problem of the greatest common 
measure for ordinary imaginary integers, was by the same author, 
and was among those partly read by me during my recent Chelten- 
ham holidays ; namely, his Sitiiations-Kalkul. About this book, 
published in 1851, I had felt a curiosity, because it professes to 
deal with the multiplication of lines in space; and I wished to know 
whether the author had in any way re-invented the quaternions, 
or made out anything better, but having the same general aim. 



* In his first rapture he exclaimed * I see that quaternions will do every- 
thing ' you will remember that these were his words, not mine. 

t In a letter to his son, written from Cheltenham, August 21, Hamilton 
writes, ' I dashed off before breakfast (indeed in my shirt) a general method [for 
its solution], though I had never thought of the problem before ; and after break- 
fast went through the numerical calculation.' 



7O Life of Sir William Rowan Hamilton. [1856. 

You are at liberty to conjecture whether I was more disappointed 
or pleased by finding that, notwithstanding his imposing symbol 

# + y v - 1 + s V* 1 v - 1> 

for a line in space, the laics of his multiplication of such lines are 
precisely those which occurred to me, in my first crude guess of 1830, 
recorded in paragraph [41] of the preface to my lectures, and by 
me very wisely discarded though Scheffler has managed to spin a 
book about them ! ' 

* From SIR W. E. HAMILTON to J. T. GRAVES, r. R. s. 

1 OBSERVATORY, September 30JA, 1856. 

4 ... And really I should be forced to consult my books for the 
purpose of even telling you the titles of the things I have written 
upon since we last met, if it were necessary that I should enume- 
rate them. 

'For I had almost forgotten, till something led me just now to 
open the large book to which I allude, that I had written about 
ten pages in it, during this expiring September of 1856, containing 
an application of quaternions to a question of solid geometry, sug- 
gested to me lately by a recent article of Grassmann's in Crelle's 
Journal \der Mathematik]. The article was one of plane geometry 
alone, but the author was well worthy to have anticipated me in 
the discovery of the quaternions; and it appears to me a very 
remarkable circumstance that he did not. See my quotation, in 
page 62 of the Preface to my Lectures, from the Preface to his 
AusdehnungslehrC) if you do not happen to have that obscure, but 
highly original, work in your library. I know that you have the 
almost equally interesting, and to me far more pleasing book, the 

of Moebius. The last-named writer is a very original one, and a 
wonih'rfnlly candid one : if you saw his letter of last autumn, in 
which he tcctiu-ul upon my theory of the circular hodograph, you 



Probably this blank space should be filled by the title of A. F. Moebius's 
book Der barycentrische Calcul (8vo: Leipzig, 1827). 



AETAT. 51.] Honesty in Science. 71 

could prove to Mrs. Graves that scientific men are not always 
jealous of each other.*' 

As has been already recorded, it was in letters commencing 
with the 7th of October that Hamilton communicated to Mr. J. T. 
Graves his discovery of the Icosian Calculus. The first letter of 
this series affords interesting proof of his scrupulous care to act 
with perfect fairness towards a brother mathematician regarding 
priority in arriving at important conceptions. On this account I 
here reproduce it. After adverting to his repayment of a trifling 
loan of money which his prolonged visit to Cheltenham had caused 
him to seek from his host, he continues : 

From SIR W. B.. HAMILTON to JOHN T. GRAVES, F.R.S. 

* October 7, 1856. 

* ... There is a much more interesting, if not more useful sort 
of honesty, which chiefly urges me to write to you at this moment. 
... I want to know whether you have anticipated me in the find- 
ing (for discovery may be too grand a name for it) of a new system 
of geometrically interpretable, and algebraically non-commutative 
symbols, quite different from i,j t k. If so, in the parentage of 
thought these new symbols, *, K, A, will be bound to pay a certain 
filial duty to your brain. For it has just occurred to me to re- 
member that you spoke in the Mathematical Section lately of a 
new imaginary, the nature of which you did not communicate, 
suggested to you by considerations of polyhedra. Now it was 
precisely that class of considerations which led me lately to my 
own result. That they (the polyhedra) interested me, is chiefly due 
to you ; but I am not conscious of your unexplained remark about 
a new imaginary having influenced my recent speculations ; nor, 
indeed, should I quite like to call my new symbols " imaginary." If 
you choose now to tell me what your result was, I shall not be un- 
willing to receive the information, since I have something to offer 



t In a letter dated Leipzig, August 27, 1855, Moebius writes, ' In my 
lectures and otherwise I have tried to divulge the most elegant and simple way, 
by which in the last-named Paper you derive the Keplerian laws from 
Newton's law of gravitation.' Manuscript Book L. 1847, p. 156. 



72 Life of Sir William Rowan Hamilton. [1856 

in return. On my part, I am about to enclose a memorandum of 
the nature of my new system, which I shall seal up in a separate 
cover, so as to give you the option of not reading it, should it be 
any satisfaction to you not to do so till after your writing to me. 
But of course I have not the least objection to your opening the 
enclosed packet at once. . . .' * 

From the SAME to the SAME. 

' OBSERVATORY, October 17, 1856. 

* MY DEAR JOHN GRAVES The constant friendship to which you 
have admitted me, during a period of more than thirty years, may 
cause you to remember, with some interest, that you were the first 

* The enclosed memorandum states with such conciseness the substance of 
the discovery that I insert it here, supposing that it may be welcome to some 
mathematical reader. 

MEMORANDUM for John T. Graves, Esq., from his friend, the Author. 

' It has lately occurred to me that several investigations respecting poly- 
hedra may be assisted by the introduction of a system of new symbols, i, , A, 
which shall be symbolical roots of positive unity, but uncommutative (although 
associative) as factors among themselves. For example, I find that if we 
assume 

i' = l, /c = l, A = l, A = i; (A) 

then the only cyclical mode of passing over all the successive faces of the 
icosahedron, or (along 20 of the 30 edges) from corner to corner of the dode- 
cahedron, is fully expressed by the formula 

Ai 8 *2 3 (*i A*) 2 = 1, (B) 

where k\ = IK, 2 = we'-. (C) 

By making, similarly, 

li = i\, fe = \* f / 3 = iA s , / 4 = <A; (D) 

so that A-i = A, and I K; (E) 

while k t and h h / have the significations stated above, I find that all results 
respecting passages over successive faces of the dodecahedron, or along su 
sive edges of the icosahedron, so as to cover a//, or some, of the faces of the one 
solid, or to pass over all or some of the corners of the other, admit of being ex- 
pressed by formula) of the same general kind as (B), with Ts instead of A's for 
factors. For other factors I use other exponents. 

W. R. II. 

' OHM i.v \ [OH "i 'I 1 . <'. !>., 
\- 



. 51.] Kirkman on Polyhcdra. 73 

person to whom I communicated my invention of the Quaternions, 
in a letter of October 17, 1843, which has since, by your permis- 
sion, been printed.* 

* I had some hope that you might have anticipated me in a 
more recent mathematical conception, the result of a train of 
thought for the suggestion of which I am indebted to yourself. 
With your usual candour you have informed me that you have no 
claim to make in the matter. Let me at least have the pleasure of 
in some degree associating your name therewith if this letter shall 
be thought worth preserving ; for I feel sure that if you had not 
lately pressed on my attention the geometrical interest of the poly- 
hedra, although the feeling of such an interest is among my very 
earliest mathematical recollections, I should not have been conducted 
to that novelf system of symbols, respecting which I have now the 
pleasure of giving you some enlarged particulars, without pretend- 
ing to attach more than a very moderate degree of importance 
to the results. . . . [Here follows a full statement extending to 
twenty-three folio pages of manuscript.] ' 

Subsequently, in a letter which I now insert, Hamilton refers 
to the investigations of the Rev. T. P. Kirkman on polyhedra. 

Fro?n the SAME to the SAME. 
[FROM A COPY.] 

4 OBSERVATORY, November 1, 1856. 

* I am much obliged to you for drawing my attention to Mr. 
Kirkman's researches on Polyedra you see that I consent to drop 
the"h." 

* Though I admire his inventive powers, I happened to be en- 
tirely unacquainted with what he had done on the subject referred 
to. I suppose that I must have just seen his name printed in con- 
nexion with polyedra ; but if I did so in the notices of the Pro- 
ceedings of the Royal Society, for instance it made no impression 
on me whatever, because I had no thought of pursuing the subject 



* See supra, vol. ii., p. 463. 

t 'Of course when I say " novel," I can only mean that it is such to myself 
at present.' 



74 Life of Sir William Rowan Hamilton. [1856. 

myself, and did not hope to understand the abridged account, 
without having the fuller statement before me. 

* It was only yesterday that I observed that the First Part of 
the Philosophical Transactions for 1856 had arrived here, and that 
I began to cut the leaves. After a fashion, I have read the two 
papers of Kirkman at the end ; enough to catch what he is at, and 
to see that on the geometrical side he has been dealing with far 
greater generalities than myself. But I do not observe the slightest 
trace of his having caught even the notion of my algorithm or 
calculus, such as it is ; and my long letter to you contains only 
specimens of such a calculus. Even as regards my " q agon," 
namely, a certain 20 agon, composed of 20 successive edges, out 
of the 30, of the ancient dodecaedron, I do not see that Kirkman 
has printed an anticipation; though I am most willing to believe 
that he possessed one, and thought it too easy to publish. 

* The conditions of his Theorem A, page 413 of the Part [of 
the Philosophical Transactions^ above cited, do not seem to include 
my result ; for though every " crown-summit " of the old dode- 
caedron adjoins one of the " wall- summits," it is not true, con- 
versely, that every wall-summit is contiguous to a crown one. 
But Kirkman, in page 418, expresses himself as aware that a 
" q agon " can be traced along selected edges of a " q acron," 
without all the conditions of his Theorem A being fulfilled. Still, 
I do not see that he has specially perceived the closed polygon 
which winds round all the twenty corners of the Dodecaedron of 
Euclid and Plato.'* 

From JOHN T. GRAVES to SIR W. E. HAMILTON. 

< CHELTENHAM, November 13, 1856. 

1 ... It is pleasant to me to find that you have not been at 
all anticipated by Kirkman, and the conception of your [Icosiau] 
calculus strikes me as not only original, but likely to lead to a 
good deal/ 

* Plato, in the part of the Timatis to which I lately referred, seems to 
epeak of tin.- l)Ml-r:u-lnn, or ratlin- In hint ut it, us the most mysterious of the 
solids, and as perhaps typifying some other world to us unknown a sort of 
quintettential /<>,!. At least such is my impression, unassisted by the writ- 
ings of any commentator.' 



IETAT. 51.] ^Letters on Quaternions? 75 



CIIAPTEE XLII. 

' LETTERS ON QUATERNIONS/ AUBREY DE VERE's ' MAY CAROLS.' 
BRITISH ASSOCIATION IN DUBLIN. CORRESPONDENCE WITH 
SALMON. 

(1857.) 

ONE of Hamilton's neighbours living at Farmleigh, a country- 
house adjacent to the Phoenix Park, was Mrs. Smythe, an English 
lady of much intelligence and culture. His visits to her house 
were often sought for, and to congenial guests there assembled he 
not unfrequently gave delight by reciting or reading to them 
poems, not confined to his own and his sister's (though these were 
in request), and by imparting to them information on various 
sciences, and discussing with them questions on connected diffi- 
culties. Linnseus and Botany, the Bible and Geology, Burns 
and Tennyson, I find by his note-books were subjects so discussed, 
and I give at foot in substance two of these memoranda, which 
are not without interest.* 

But I here mention this lady because, by seeking from Hamil- 
ton on behalf of a sister (Mrs. Frederick White) whose tastes were 
scientific, a general notion of Quaternions at the same time that 
Dr. Nichol of Glasgow was urging him to contribute an article on 
the subject to his Cyclopaedia of Science, she was the cause of this 

* ' Is it not Bacon who says that truth will more easily emerge from error 
than from confusion? Linnaeus, it seems tome, considered himself as acting 
on that view, and regarded his confessedly artificial system as only the pioneer 
to some more natural system of Botany (for Linnrcus's system was /</// natural 
in many respects) such as that of Jussieu has been accounted to be, and whirh 
in its turn may, and probably will, merge in some still more natural - 
If I even dimly remember the Latin words of Linnaeus as printed in a delight- 
ful little book not very generally known even to botanists, the great Swedish 



76 Life of Sir William Rowan Hamilton. [1857. 

article taking the form of three Letters on Quaternions, the first of 

which was addressed to " Mrs. S ." He felt, however, that 

what he had to write in addition to the contents of that first letter 
could not with probability be supposed to be intelligible to a lady 
who was no more than a dilettante in science, and on this account 
the second and third letters were addressed to an imaginary 
gentleman. They give a very full view of the nature and modes 
f application of the Quaternion Calculus, and of his own work 
upon it up to the time at which they were written : and in them, 
as in all his writings, he takes care to assign to other mathema- 
ticians engaged in the same field of research their due amount of 
credit. The conclusion of the third letter gives an interesting 
statement of his views and feelings with regard to his own re- 
lation to his Calculus, and of his expectations as to its future. 
His estimate of what he had accomplished in the use of it as 
an instrument is most modest : the last few sentences of the 
letter may here be reproduced : 

' OBSERVA.TOKY OF TRINITY COLLEGE, DUBLIN, 
January 24, 1857. 

' . . . The quaternions seem to me to admit of entering into an 
alliance so close, yet new, with every part of pure and applied geo- 
metry, and at the same time to require such large additional develop- 
ments, before their relations of analogy and contrast to existing 
methods of calculation shall be fully known, that I count myself 
) m' rely to have begun them. The field is far too wide to be tilled by a 
solitary labourer, even with occasional assistance from a few friends, 



naturalist did say " Primum atque ultimum in Botanica methodus naturalis 

1 1 suppose that we must admit some symbolical element into the inter- 
pretation of the First Chapter of Genesis. At least the obvious and literal 
interpretation of it appears to be no more consistent with known facts of science 
than those passages which speak of the Earth as being at rest, &c., in other 
parts of the Bible . . . What right or reason have we to imagine that ;i 
scientific revelation \\ -.^ intrust. <1 to Moses? and for the Jews! those children 
to whom tin Law was as a schoolmaster; though 1 know that to them were 
committed in a most high and sacred sense the "Oracles of God." ' 



AKTAT. 51.] Honesly. I'ani/y. 77 

who feel some interest in his exertions. The time may come, 
though, if so, it will be due to other explorers rather than to me, 
when, the mat hematic* of this calculus having become comparatively 
mature, it shall admit of being extensively and usefully appl'ml 
to y>////.s/rx, as a new instrument in the study of Nature. In Ili 
prospect of such a time, I feel with no jealous pain, that although 
it may have been permitted to me to accomplish somct/thif/ in 
this enterprise as an honourable Suitor of Science, yet the Bow 
awaits its Ulysses.' 

Early in the summer of 1857 an additional and striking pro >f 
was afforded of Hamilton's scrupulous care to guard the rights of 
others as well as his own in scientific discovery. His friend Mr. 
John Graves had sent him from Cheltenham, for communication 
to the Royal Irish Academy, a Paper entitled'^. Fundamental 
Theorem respecting Congruences affecting a Class of Complex Integers 
which invoice the Imaginary Cube Roots of Unity* : having received 
permission from the Council, Hamilton had given formal notice of 
the reading of this Paper, carefully assigning the authorship to 
Mr. Graves : but by some mistake this notice, when printed, 
omitted the clause naming the author, and thus the Paper appeared 
to be a contribution from Hamilton himself. He instantly, at his 
own expense, had 500 copies of a memorandum correcting the 
mistake printed for distribution among the members of the 
Academy : he did this, not thinking it right to wait for the next 
Academy meeting, at which in point of fact, he read the Paper, in 
conjunction with two of his own, but taking particular care to 
indicate the distinct authorship. It may not be unnecessary to 
point to the fact that the Paper was one of which he was willing 
to be the sponsor, and the results of which he employed himself in 
extending by means of Quaternions. [See MS. Book, 1856. 4to.] 

As another indication of character, I insert a note written in 
the summer of this year to the same old friend. The foible of 
vanity was one from which he felt that he was not exempt. In 
this note he ingenuously betrays his consciousness of the infirmity, 
the evil of which he seeks not to palliate ; and he takes a legit i- 



78 Life of Sir William Rowan Hamilton. [1857. 

mate pleasure in what he rightly judged to he a proof that it did 
not deeply penetrate his nature. The letter from Mr. J. T. Graves, 
to which this was a reply, had brought to his knowledge the glow- 
ing recognition of the discovery of Quaternions, recently pub- 
lished in the North American Review* and at the same time had 
made mention of the illness of the writer's sister, Madame Eanke. 

1 OBSERVATORY, July 26^, 1857. 

* MY DEAR JOHN GRAVES I remember reading, long agro, in 
Coleridge's Biographia Literaria, or in some other work of his, a 
passage which ran somewhat thus : " For the passion of Vanity 
is a loveless passion loud on the hustings, gay in the ball-room, 
mute and sullen by the family fireside." I hope that it may be 
accepted as some indication of what, I trust, is the fact, that my 
moral nature has not been hopelessly corroded by vanity, when I 
tell you that your slight mention (though made with all due 
fraternal affection, and, therefore, not deliberately to be called 
slight] of your sister so entirely obliterated all recollection of the 
compliment which you say that I have been lately paid in 
America about the Quaternions, that I had absolutely forgotten 
all that part of your recent letter when I was lately writing to 
you. Yet I remember, with great interest, that my scientific 
labours were very early noticed in America. But, somehow, I 
look quite coldly now on compliments of all sorts, though certainly 
not less genial in temperament, nor more indolent in spirit and 
exertion, than I was long ago. Here is a whole basket of egotism 
to prove that I am not egotistic ! which, after all, I am, but still 
not so much as to obscure some common sense, nor to deaden any 
affection. Affectionately yours, at all events, 

W. E. H.' 



The following portion of the correspondence with Mr. De Vere 
will be welcome to the reader, as casting additional light upon the 
mutual relations of the two friends : 



I, vol. ii., p. 455. 



AKTAT. 51.] ^May Carols? 79 

From AUBREY DE YERE to SIR W. E. HAMILTON. 

' 7 PARK-STREET, WESTMINSTER, July 29, 1857, 

' On my arrival here, a few days since, I was greeted by a 
letter from you a little more than a year old ! It was dated July 
18th, 1856, and related in a large part to Mr. Penny, for your 
kindness to whom pray accept my best thanks. ... I only wish 
I had had an opportunity of making you and Dr. Newman 
mutually acquainted. However much you may differ on some 
subjects, there are a hundred which would alike interest you both ; 
and that two such persons should have been so long near each 
other, without even meeting, is a piece of bizarre irony on the 
part of that Social Fate which holds us in her iron meshes. But 
you both seemed full of occupations and engagements, and pro- 
bably had no time for new ones. He is now at Birmingham. 

1 1 have had a charming winter and spring at Home. All the 
days of both seasons passed in a satisfactory way, except one sad 
fortnight spent in the dying room of my poor friend, Eobert 
Wilberforce (late Archdeacon Wilberf orce) . It really seems as if 
Eonie were inexhaustible. Every year one has to begin again, 
and open on a new world of interest. On my way back I visited 
several marvellous old Etruscan cities, in the neighbourhood of 
which are tombs 3000 years old, stored with all manner of things, 
from monuments to ear-rings, as fresh as the day they were made. 
At Venice I passed three weeks, sailing about the water-streets of 
that Queen of Cities the Cleopatra of the waves, and the most 
beautiful of earthly pageants. On my way home I sojourned for 
a few weeks beside the Italian lakes, and under the shadow of 
Mont Blanc ; and now here I am again in Babylon. 

1 1 do not know whether you have or have not received a copy 
of a new volume of poetry by me, called Mai/ Carols ; but I am 
pretty sure I directed Longman to send you one. It is an attempt, 
like my hymn on the Feast of St. Peter's Chair at Rome, to put 
theology into verse ; but, to make it less dry, I have broken the 
poem into a number of short pieces, thus making it a serial poem, 
such as were common in the Elizabethan age, as well as in Italian 
literature a form of composition revived in our time by Words- 
worth (Sonnets to Liberty), and Tennyson (In Mtmoriam), and 
others. I have endeavoured in each piece to indicate, as briefly as 



8o Life of Sir William Rowan Hamilton. [1857. 

possible and in mere outline, some one leading idea or principle bear- 
ing on the Blessed Virgin, and therefore illustrating that vast and 
wondrous doctrine the Incarnation which in these days is so 
little really taken in. The position assigned to Mary in the Scheme 
of Redemption is a theology in itself ; and, without mastering it, I 
own it seems to me most difficult to master with the mind, as well 
as realize with the heart, the Divine Economy of the Incarnation, 
and of Mediation. I have not, you will observe, shrunk from 
making statements as strong, doctrinally (apart from exaggerated 
or rhetorical language], as have ever^been made ; but, on the other 
hand, I have endeavoured, in other pieces, to affirm with equal 
plainness those other great, and, as I think, correlative Christian 
Doctrines, which, in the eyes of many Protestants, appear incom- 
patible with the assigning of so exalted a place to Mary. In 
other words, I have endeavoured not to extenuate the " Cultus 
Deiparoo " and present it in a flattering form, but to illustrate it 
at both sides ; not drawing a " Via Media " line, but drawing a 
circle large enough to include the converse Truths (the one set 
belonging to pure Theism, and the other to the Incarnation), 
which must be reconciled in the theology of the finite creature 
elected to be the Mother, in His Humanity, of her Infinite Creator. 
I do not know whether the book is at all calculated to recommend 
the Roman Catholic teaching on this subject ; but I should think 
it may help to make it more intelligible to minds willing to regard 
the subject with candour, and with the attention required by a 
subject at once so large and so subtle as Theology. At the same 
time, I should think it not unlikely the poem may somewhat 
frighten those who have been in the habit of looking at the sub- 
ject only from one point of view, and with the associations that 
have gathered round the Protestant tradition. You must tell me 
candidly what you think of it, both philosophically and as a 
literary experiment. I send you a copy. If you have got one 
already, pray transfer it, with my compliments, to your frieiid 
Mrs. Wilde/ 

From SIR W. R. HAMILTON to AUBREY DE VERB. 

1 OBSERVATORY, July 30, 1857. 

* MY DEAR AUBREY DE VERE We have not written to 
other for a long time at least a year ; but you have often saic 



. si.] A Tetrad. 81 



what I always recognised to be the truth, that you and I, on every 
occasion of fresh writing, went on precisely as if there had he^n no 
gap in our correspondence. In fact my love to you has never been 
for a moment impaired nor my respect ; although I am farther 
than ever from participating in your Homeward movement : I do 
not call it change ; for, on looking back on those old days, when 
we seemed so much to agree, not only in sentiment, but also in 
belief, on almost every subject, I can see that, while you were (in 
heart and I do not mean in disguise) a Catholic aiming to be a 
Protestant, because your father was so, I was throughout essentially 
a Protestant, though with many Catholic sympathies. I request 
you to believe that I do not belong to the class of irreverent 
scoffers at Popery.' 

From, SIR "W. R. HAMILTON to MRS. WILDE. 

' OBSERVATORY, April 13, 1858. 

' Can I better, or more pleasantly, commemorate our first 
meeting, which occurred exactly three years ago, on the 13th of 
April, 1855, at the hospitable house of Colonel and Mrs. Larcom, 
than by introducing to you my old and dear (I regret that I am 
obliged to add my Popish) friend, Aubrey De Vere, the poet and 
prose- writer, with some of whose principal works in prose you are 
already acquainted ? 

* Some faint remonstrance notwithstanding, the modern Roman- 
ism, of which Aubrey is a splendid type, tends to the introduction 
of a Tetrad into the human notion or belief of the Divine Nature 
or Essence. 

* I cannot believe that this result, or even this tendency, is at 
all short of idolatry. 

* It is idle to talk of Images exposed, and falsely pretended to 
talk or wink : the Poems of Aubrey De Vere sufficiently exhibit 
the essential idolatry (for the word " Mariolatry," though just, 
may be dispensed with as unnecessary) of the Church which he 
has adopted. 

* I did not intend to say so much : in general, I care very little 
whether an acquaintance of mine is of this or that religion ; and, 
in the present case, I retain a very deep affection for the indivi- 
dual author and friend of whom I have been writing. 

* He will never be a Protestant again ; or, to speak more 

VOL. III. G 



82 Life of Sir William Rowan Hamilton. [1857. 

accurately, I venture to pronounce, from what I knew of his boy- 
hood, that he never MY/.V a Protestant : although, of course, we are 
bound to believe that he thought himself to be such, when he wrote 
that grand essay on English 3Ii*ni/e and Irish Misdeeds, of which 
you lately accepted the loan, and which you joined me in ad- 
miring.' 

The last week of August, in 1857, brought to Dublin the 
members of the British Association for the Advancement of 
Science for their second meeting in the Irish metropolis. It will 
be remembered how distinguished a part was taken by Hamilton 
at the first meeting in 1835, when he acted as joint- secretary with 
Professor Lloyd, under the Presidency of the venerable Provost, 
Dr. Bartholomew Lloyd. Professor Lloyd had now become Dr. 
Humphrey Lloyd, one of the Senior Fellows of Trinity College, 
and had been chosen to be President for the year. Lloyd's ex- 
pressed wish had been that this honour should be conferred on 
Hamilton ; but it was thought that his own claim was one which 
ought not to be set aside, and the reader will recall that at 
Cheltenham, at the meeting of the previous year, Hamilton had 
seconded the proposal that his friend should be appointed to the 
office. As a Vice- President, Hamilton was a regular attendant at 
the meetings, and he made a communication to the Mathematical 
Section, On some Applications of Quaternions to Cones of the Third 
Degree. This application of his Calculus to Surfaces of the Third 
Degree was considered by Hamilton to be in the working of Qua- 
ternions a step forward of very considerable importance. This is i 
proved by a letter written on the 14th of April in this year to Dr. 
Lloyd, which for that reason I here insert : 

* One thing that has delayed me has been, as you may be gli 
to know, my having had occasion to write to Mr. Salmon, 
announce to him a totally new application of the Quateniim 
namely, one to Cones (and in fact to Surfaces) of the Third Order. 
Strong as my d priori confidence in the Quaternions has been fc 
many years, on account of the laborious investigation to whirh 
have long submitted their principles, I still have had a misgiving, 



. oi.] Cones of the Third Degree* Henry Smith* 83 



their ftriictical uw fulness might be bounded within the (not 
narrow) limits of the theory of Surfaces of the Second <)r<l<r. At 
all events, I was content for many years to limit /y///W/'to the study 
of such Surfaces, as objects for the application of the Quaternions. 
But it occurred to me, as I was walking up through the fields last 
night, after attending a meeting of the lloyal Irish Academy, 
that I could represent the General Cone of the Third Degree by the 
very simple binomial equation, interpreted on the principles of my 
Calculus, 

^O; (A) 



where p is the variable rector, or side of the cone, and q, q', q' are 
three arbitrary but constant quaternions. Three Important planes, 
analogous to the two real cyclic planes of a cone of the second order, 
are thus at once suggested to me, namely, those denoted by the 
three following equations. 

Sqp = 0, Sq'p = 0, Sq"p = 0. (B) 

And a fourth plane (C), which is a sort of transversal to these 
three, has been obtained by me this morning through calculations 
sufficiently easy ; but which plane, with the former, seems likely 
to play an important part hereafter. I am ashamed to say that I 
have never read Newton's Emnneratio, &c. Now that I have 
caught hold of a single formula (A), which must include all old 
results of that sort, I long to sit at his feet. . . . '* 

The AtJienmim (September 12, 1857, p. 1148), recording the 
above communication to the Association, adds : 

* It would be impossible to give the general reader any clear 
idea of this abstruse paper; but at its conclusion the soundness of 
the principles on which the author proceeded was made strikingly 
manifest to the Section by Mr. Henry J. Smith of Oxford, 
explaining in fully as lucid a manner as that of Sir W. Hamilton 
(who makes everyone that hears him for the moment think that 
he clearly comprehends the whole subject), how by the method of 
Quaternions, but by a different process from that of Sir "W. Hamil- 
ton, he had in some of the examples selected by Sir W. Hamilton, 
arrived at precisely the same numerical results.' 

* Compare Elements of Quaternions, 1866., p. 705. 
G 2 



84 Life of Sir William Rowan Hamilton. [1857. 

By a letter to Mr. Salmon, of August 30, 1857, given further 
on, it will be seen that Hamilton, in his characteristic way, free 
from all jealousy and welcoming co-operation, was anxious to 
secure for Henry Smith an opportunity of communicating his 
results in full to the meeting. The impression made by this ex- 
cellent mathematician, accomplished scholar, and public- spirited 
citizen upon his countrymen (for though an alumnus of Oxford 
he was a native of Ireland) was in all respects at one with that 
which, wherever he was known, made him the delight and admira- 
tion of his friends. The intercourse between him and Hamilton 
did not cease with this meeting ; I find that in after days letters 
passed between them. 

Hamilton also expounded to the Mathematical Section his 
Icosian Calculus, and distributed a lithographed illustration of the 
Icosian Grame) And he made an oral communication in connexion 
with Quaternions, for which I find some incomplete preparatory 
notes, which seem to me to contain points of importance in the 
history of the development of that Calculus : I therefore insert 
them here. 

* August 29^7?, 1857. ... As I shall have occasion in my 
communication of this morning to introduce a few remarks on 
geometrical and symbolical imaginaries, I may say here that I 
have long ceased to call my symbols i, j,k, imaginaries. They are 
indeed square roots of negative unity ^ and are so far analogous, in 
calculation, to the old ^/-l. But besides those other differences, 
even in calculation, on which I have often remarked, and especially 
those which arise from my non-commutative law of multiplication, 
when I came to see how very easily and simply the i, j, k, may be 
conxtrm'tetl in geometry, as denoting a system of three rectangular 
unit tines, I soon abandoned the name imaginary, as relative to 
i, j, k ; and I felt, for some time, a hope that we might thus come 
to get rid entirely of itnaginaries, and might retain only reals, 
in calculations thus applied to geometry. But just as in man 
well-known researches of modern geometry I need scarcely men- 
tion till- c 1. l.inted writings of Chasles nor need I speak of the 
works of our own Salmon intaijinary jtoints, linos, and surfaces, 



. 52.] Imaginarits. LctU> a/ J//-\. Ui>\'d. 85 

are boldly and copiously and usefully employed ; so in the pro- 
secution of quaternion*, and in their continued application to 
y<'onn'try, I found it absolutely nrn-wtry to admit geometrical 
riex, and to represent them by symbols, in which the old 
and ordinary \/~l was combined with my own i, j, k, and which I 
called bi vectors and biquaternions.' 

On the last day of the Association week, although he had to 
attend in the morning the funeral of a relation, he made a special 
effort to be present at the final meeting, in order that he might 
manifest to the end his loyal homage to the Presidency of his 
friend. For this loyalty he could not have received a more 
gratifying reward than was conveyed in the earnest thanks ex- 
pressed to him at the time by the wife of the President ; and in 
the letter, written by her shortly afterwards, which, without 
asking her permission, I venture here to insert. No one who has 
known the writer what she is in herself, and what she was to her 
husband will be surprised at its high feeling and noble expres- 
sion. Hamilton, with his habitual sense of justice, forwarded the 
letter to Mrs. Robinson, in order that her husband might enjoy 
his coequal share in this generous recognition of generous feelings. 



From MRS. HUMPHREY LLOYD to SIR W. R. HAMILTON. 

'KiLCEONEr, September 15, 1857. 

*MY DEAR SIR WILLIAM Having at last an hour of leisure, 
after the many remnants of business which remained to be arranged 
previous to our leaving home for a few weeks, I wish to send you 
a few lines, that you should no longer marvel at the warmth with 
which I spoke on the day of the closing meeting : as indeed I do 
not wonder that you should have hitherto done, for I dare say I 
expressed myself with an energy which is very unusual with me. 
I was, at the moment, under the influence of many feelings and 
recollections which very deeply moved me ; and in the words I 
uttered were in fact contained the thoughts which reached back to 
former times and to other persons as well as yourself. I had seen 
on that day, and throughout the meeting, Dr. Robinson and your- 



86 Life of Sir William Rowan Hamilton. [1857. 

self (many others I might also name) rejoicing at, and assisting in, 
all that was to bring honour to Humphrey. I knew well that this 
cordial feeling, however generous it might be, was nothing uncom- 
mon, but that it had always been thus ; and knowing as I do how 
rare is such feeling how little the most exalted intellectual merit 
exempts from a lurking jealousy, when the claims of another are 
in any degree brought forward I thought it was no less beautiful 
than to me most delightful to mark the warm appreciation of my 
husband's character, moral and intellectual, which you then an<l 
always seem to take pleasure in showing. I felt, too, how very 
gratifying it is that he has been so little entangled in any angry 
strife with his brethren, and how precious was the kindly sym- 
pathy of those amongst them who had known him best and longest. 
Thus you will perceive that my words to you were the expression 
of a complicated emotion, concentrated upon you as the representa- 
tive of so much which called for my gratitude. I am writing in 
haste, for I wish to send these few lines before I leave home ; but 
I think you will not be at a loss to understand what I mean to 
convey, nor will you deem the feelings which influenced me either 
misplaced or mistaken. Believe me sincerely yours, 

6 DORA LLOYD.' 



Early in the May of 1857 Hamilton entered upon a correspon- 
dence on Quaternions with Dr. Salmon, at that time Donegal 
Lecturer in Mathematics in the University of Dublin, and no less 
eminent then for his mathematical than he has since become, as 
Regius Professor of Divinity, for his theological works. In this 
correspondence, which was carried on to the end of September, 
Dr. Salmon places himself as a learner at the feet of Hamilton, 
states frankly his difficulties and objections, gradually overcomes 
his original repugnance to the revolutionary character of some of 
the processes, and while retaining a doubt whether a large part of 
the working equations in Quaternions might not admit of being 
conveniently translated out of the quaternion notation, concludes 
by saying (August 26th, 1857) that, * even if this were so, the 
admirable consistency and harmony of the whole scheme is de- j 
serving of all praise.' 



AI-TAT. 52.] To Dr. Salmon on Quaternions. 87 

In reference to these words of Dr. Salmon, it may be here 
suggested that the subsequent history of Quaternions has esta- 
blished, at least in certain departments of research, the practical 
advantages of the Calculus, as well as its theoretical perfection; 
indeed, this result is anticipated by Dr. Salmon himself ; for an 
earlier letter of his (June 24th, 1857) closes with the following 
sentence : ' I hope you will not be lazy about the Covariants and 
Contravariants. I have no doubt that great fruits will come from 
the marriage of the Quaternions with the Calculus of Forms, and 
no one but you ought to give away the bride.' 

This correspondence between Salmon and Hamilton is one 
which, would necessarily prove of the highest interest to students 
of Quaternions, who would find in it the real difficulties of the 
Calculus urged by the ablest of learners, and solved by a teacher 
of all others the most competent. It is to be hoped, therefore, 
that in due time it may be transferred to print. I make here a 
few extracts touching points of interest, from letters of Hamilton 
to Dr. Salmon and Dr. Ingram. 

From LETTERS to DR. SALMON. 

[A. 1857, p. 415] June 23, 1857.*. . . It is fair to say that 
(when too late) I found that Grassmann had independently (but 
perhaps not quite so soon still it is not a matter worth contesting) 
arrived at the same conception and notation, respecting the DIF- 
FERENCE OF TWO POINTS (B - A), regarded as their directed dis- 
tance what he calls " strecke," and I " vector." But this is merely 
a preparation for quaternions, and not as yet in amj degree the Doc- 
trine of the Quaternions themselves. 

' I admire Moebius very much indeed, but he has (I think in 
his Barycentric Calculus, &c.) approached lew nearly to the quater- 
nions than Grassmann in his Au-sdehnungsleht'e' 

' July 25, 1857. No copy was preserved of the note, and 
indeed I hope that nobody will be so unkind to my memory as to 
dream of printing all the (perhaps many thousands of) notes or 
letters which I have written in the course of my life ; though I 



88 Life of Sir William Rowan Hamilton. [1857. 



am not quite so modest as not to fancy that a judicious and 

or, if you like the word better, unsparing selection might yield 

something which " the world would not willingly let die." In short, 

destroy at least 90 per cent., and the last tenth might be perhaps 

preserved. 7 

[A. 1857, p. 481]. * P. S. I wonder whether Cayley (or 
indeed any one else) is as yet able to read my Appendix C. At 
this moment I do not believe it. 

* I see that my P. S. has not been a particularly modest one ; but 
can assure you that Cayley has written heaps of things, which / 
could not without deep study read.' 

[A. 1857, p. 483]. July 27, 1857.*. . . Observe for a moment 
the vast range of the aim of what my brother astronomer Airy, not 
usually supposed to be a flatterer, was pleased to call, on receiving 
my book, the "large science of Quaternions." The question is not 
what Hamilton may be able to make of it, but what his conception 
of the matter is, and what he hopes that others will yet accomplish. 
And the AIM is no less than this : to impregnate all existing algebra 
(including the Differential and Integral Calculus, Calculus of 
Variations, &c.), and all pure and applied geometry, with a NEW 
ELEMENT of thought and calculation (through ',/, k, &c.). Algebra 
(including the covariants) becomes thus in thought I am very far 
from saying in achievement that particular CASE of the quaternions 
for which the coefficients of i,j 9 k, are zero. But you are well aware 
that the memory must be trained, in any department of study 
whatever, before results and rules can become ready to be applied. 
Still I do not despair of classifying the chief rules and results which 
I myself habitually, and almost unconsciously, employ (all ulti- 
mate I ij derived from the laws of t, /, k, or from the formula, 
t* =f = k* - ij k = - 1) so as to assist the memory of a student 
and even to draw up some very SHORT manual, which might not 
task too much the patience of an undergraduate. At present I 
know that my book is quito unfit to enter into the undergraduate 



[A. l*.">7, ].. /il">|. A iHjitatU, 1857. 'Meanwhile lam charm.-.! 
by your saying that i/* your result does not agreu with mine, so 
much the wor.*< I'm- me! 



AETAT. 52.] Letter to Dr. In gram. 89 

* That is the true and rational footing to put the thing on ; and 
far more really flattering, or pleasing to me, than if a person were 
to say as there is a story about a courtier saying to Marie 
Antoinette, in those days when (Edmund Burke) " a thousand 
swords would have leaped from their scabbards " to avenge the 
slightest insult to her, and when she is reported to have inquired 
wimt o'clock was it ? " Whatever your Majesty pleases ! " 

* Mine is a constitutional government and more, in it the king 
can do wrong. I admit this the more cheerfully, because as yet I 
am not aware that you have caught me tripping. But you may do 
so ; and I trust that when the time comes you will find my good 
humour invincible. 

4 Go on, by all means, and don't fancy that I want to teach 
people conies instead of quaternions. I only thought that cones 
might be selected as a sort of basis of illustrations/ 

From LETTER to DR. INGRAM. 

[A. 1857, p. 517]. August 14th, 1857. * Salmon is getting 
on so awfully fast in the Quaternions, that if I don't take care 
we shall get into some contest of priority! 

* In a note received from him this morning he says at the end 
of some railway pencilling "I have not got your Paper to see 
whether this agrees with your result, but if it does not, so much 
the worse for you!" You conceive, of course, that this is just 
the thing I like. I am quite tired of being a Fee-faw-fum, in 
Quaternions, or in anything else. The highest reward that can be 
given me for my labours in that, or in any other department of 
science, is to take it out of my own hands. 

* As regards my old researches in Physical Astronomy, such a 
result has already taken place, to my great satisfaction. Jacobi, 
very early, did me the honour to take them up as a commentator : 
and, of course, as an enlarger nil fere non tetigit ; nil tetigit 
quod non ornavit. More latterly, Dr. Houel, whom the Com- 
mittee of the British Association intend to invite to Dublin, has 
published two Theses (running to a large number of quarto pages) 
in Paris, for his degree as "Docteur es Sciences," on Mechanics 
and on Astronomy, almost entirely relating to what he is pleased 
to call my " discoveries." In a lithograph dated Caen, November 



go Life of Sir William Roivan Hamilton. [1857. 

8th, 1856, and entitled "Note sur le Theoreme d' Hamilton et de 
Jacobi, et sur son application a la theorie des perturbations plain'- 
taires," Houel says : " Mon but, en redigeant cette Note, a ete de 
faire voir combien, de toutes les methodes qui ont ete propo 
pour arriver aux Equations de la variation des constantes arbi- 
traires, celle que Jacobi a deduite des decouvertes d' Hamilton est 
la plus directe et la plus simple." 

* It is a genuine pleasure to me to believe that in Salmon I 
shall have a worthy successor, as regards my more purely mathe- 
matical researches, and may he much excel, even in quaternions, 
myself! Meanwhile I am,' &c. 

' P. S. Mac Cullagh once said to me, in allusion to you, that 
it was no wonder that J. K. I. should have a friendly feeling 
towards ij k. y 

P. S. OF A LETTER to DR. SALMON. 

[A. 1857, p. 518]. August 15th, 1857. < By this time you 
are prepared to admit, I think, these three things at least about 
the calculus of Quaternions : 

* I. that it is not in any slavish way dependent on co- 

ordinates ; 
* II. that it is remarkably rich in transformations; and, 

* III. that it is no mere calculus of rotations ; 

but involves a reference to lengths, translations, and (in general) to 
extensions, as essentially, though perhaps not quite so characteristi- 
cally, as to directions. 

'Your question about the two meanings of i is one of 
extreme importance. It is THE difficulty in the theory of the 
geometrical i//trr/m((ition of Quaternions: but you conceive that I 
up the result; though I must explain it further.' 



[A. 1857, p. 5:3:3]. September 15th, 1857. ' I hope you do not 
think me so />//*// as to imagine that quaternions will ercr SUJH r- 
sede <l> tinnindntx. All that I pretend is, that the two things are 
not iiU-Hticul; and that on occasions (which I may, perhaps iguo- 



AETAT. 52.] To Dr. Salmon on Quaternions, 91 

rantly, flatter myself by thinking to be numerous) the quaternion 
may have some advantage. 

'As you seem to have withdrawn, for the present, from tho 
subject, let me at least ask you to allow me to thank you for the 
kind attention which you have lately given to it. 

' And do not imagine it to be flattery, if I add that I con- 
sider you to be -very likely to excel mywlf, even in ////*, which (of 
Into years) may be regarded as my own department, if you shall 

r be induced to pursue it. " To him that hath shall be given." 
You possess (i far larger stock of geometrical knowledge, although I 
have always had something of a geometrical taste, and carried 
away all the honors of my division in geometry, when I was, very 
long ago, an undergraduate in our University. If you shall ever 
seriously take up the Quaternions, your geometry will enable you 
to go far beyond me in that subject.' 

P. S. September 22nd, 1857. * This note has been laid aside 
somewhere for a week, and during the interval I have gone 
through a frightful amount of algebraical and arithmetical calcu- 
lation, on the subject of a certain class of definite integrals which 
I do not pretend to have connected with Quaternions. So much 
the better ! You or any other friend of mine could not desire me 
to become the slave of any genie of the ring, or lamp, which may 
have once answered to my bidding. 

* In this instance I am falling back on a research begun as long 
ago as 1839. In my Paper on " Fluctuating Functions," published 
in our Irish Transactions [vol. xix., part n., p. 313.] which Paper 
interested Sir John Herschel, and was partly commented on 
by Gr. Stokes I gave a law for the. terms of a descending scries 
designed to represent the integrals 



2 fii 

da) cos (2t cos &r 

Tjo 



(I quote from memory), where t is positive and large. A hint 
from Poisson is acknowledged. Poisson had given, according to 
my recollection, a term or two of the descending series, but not the 
law, which is by no means obvious. 

[A. 1857, pp. 536, 537J. September 2-M, 1857. 'Call them 



92 Life of Sir William Rowan Hamilton. [1857. 



[Quaternions] a method of polar co-ordinates, by all means, if 
you choose. I am ashamed of having been so stiff upon that 
point. Of course you know all about lengths entering. I only 
doubted whether I had made it prominent enough/ 

[A. 1857, p. 530.] August 30M, 1857. * I thought your 
transformations very skilful, and my IMPRESSION is, that some 
(indeed all) of your forms for the single-sheeted hyperboloid were 
new to me (your Paper is not beside me at this moment). But you 
will not think me greedy of priority, if I should happen to meet 
some of them in unprinted books of mine, and if I shall venture 
to tell you so. In point of fact I am only too careless about 
priority : and yesterday, for instance, I was quite willing to admit 
to Hart that whatever Mr. H[enry] Smith's formula for the 
number of cyclic (or, as he called them, circular] planes of a cone 
of the nth order might have been, it was probably one which I had 
not anticipated. But within a few minutes afterwards I recol- 
lected that I had found that a cubic cone had fifteen cyclic planes ; 
and of course that I must have used the general formula n(2n-l), 
which I suppose was what Mr. Smith had stated in the Section. 
We ought to get him an opportunity for coming forward with 
a Paper of his own.' 

An extract from a letter, written this autumn to John Nichol, 
brings to light an old criticism of Lord Brougham which confirms 
the opinion that, great as were the intellectual powers of the 
versatile Chancellor, he was deficient in higher imaginativeness, 
and also in accuracy as to facts. It may be balanced, however, by 
a few words written by Hamilton, concerning the great speech 
which Lord Brougham delivered in 1858, at the unveiling of 
Newton's statue at Grantham that * it was worthy of the occa- 
sion.' In the erection of this statue in the native place of the 
great mathematician, Hamilton had taken zealous interest, making 
endeavours, though with little success, to enlist Irish subscribers to 
the fund raised for the purpose. 



AII \ i. 52.] Lord Brougham. Decimal Coinage. 93 

From SIR W. R. HAMILTON to JOHN NICHOL. 

OBSERVATORY, NEAR DUBLIN, October 31, 1857. 

* From childhood I have been accustomed to think of I Inly 
Eve (or All Hallows' E'en) as, to me, a poetical day, from feeling 
ami association ; and I give you my word that I have not written 
a single \ryz, nor even a poor ijk, since I awoke this morning ; 
although I plead guilty to having been caught for a while by one 
of the immortal works of Lagrange. I may have told you that 
Lord Brougham gilding the pill with praise more high than I 
could have expected was pleased to allude, unmistakably, in an 
old Edinlmryh [Renew 1 ], under the head False Taste, to an expres- 
sion of mine* which had spoken of Lagrange's great work, the 
Meann'ijuc Analytique, as a "sort of scientific poem." To be sure, 
his lordship committed, on that occasion, the slight mistake of 
confounding the said work with Laplace's Mecanique Celeste ! at 
which blunder your father would smile. But Lord Brougham is 
a fine fellow for all that ; and I think that De Morgan told me he 
had seen him reading my Quaternions in some railway carriage, or 
elsewhere. He has been lately writing to the French Institute on 
some supposed difficulties in the Integral Calculus.' 

Towards the close of this year, prompted by the receipt from 
Professor De Morgan of a pamphlet on Decimal Coinage, Hamilton 
entered with zeal into a discussion of the question, showing him- 
self to be a decided advocate of the proposed change. In the 
following spring he was officially applied to by Lord Monteagle, 
^then Chancellor of the Exchequer, for a statement of his opinion 
on the subject, and he bestowed upon it a large amount of careful 
thought. As there can be little doubt that a change in this direc- 
tion must eventually be effected, in the interest of international 
intercourse, of commercial account-keeping, and of economy of 
time, and money, and brains in national education, I think it 
right that the name of Hamilton should be allowed its weight on 



* In the first Essay on Dynamics, Philosophical Transactions, 1834, part ii., 
247 ; also in his Inaugural Address to the Il.l.A. as President. 



94 Life of Sir William Rowan Hamilton. [1857. 

the reforming side, in conjunction with those of his great scientific 
contemporaries, Herschel, Airy, Babbage, Peacock, Whewi-ll, 
and De Morgan ; and with this view I insert a letter, written by 
him to an unknown correspondent, which sets forth his views; and 
I may refer those interested in the subject to the Parliamentary 
Blue Book, in which is inserted the draft of a bill drawn up by 
him to indicate how easily, as regarded legislation, the change 
might be brought about.* He learned, however, from his friend 
De Morgan, that a change so great affecting the coinage belonged 
to the prerogative of the Crown ; and that the proper action of 
Parliament in the matter would be to address the Crown, pray- 
ing it to exercise its prerogative. 

The details of the proposed reform are fully discussed in the 
correspondence of Hamilton with De Morgan. 

From SIR W. R. HAMILTON to AN UNKNOWN CORRESPONDENT. 

* OBSEEVATOBY, October 30, 1857. 

* ... As to the Coinage, I thank you for sending me the 
printed extracts, which I intend to enclose to my friend Professor 
De Morgan. 

* The very clever writer is, as you are probably aware, quite 
on the opposite tack to my own : he is what he himself calls a 
"Little-endian," while I am decidedly, in the same mode of 
speaking (not by any means invented by myself), a " Big-endian." 
He wishes to go up from the Penny ; I wish to go down from the 
Pound. This question must, of course, remain an open one until 
an Act of Parliament shall close it. But, in the meantime, I must 
protest against the notion that 6000 tons of our present copper 
money, or even so much as a single ounce thereof, must of neces- 
sity be withdrawn from circulation if the Pound and Mil system 
shall be adopted. It will be only necessary to declare that hence- 
forward the coin, a " Penny," shall be legally the " thousandth 
part of JL Tumid." Not a ximjlc ncic coin, nor anyone ncir word, 



* Sessional Papers of 1857, Sess. 2, vol. \i\., Questions communicated by 
Lord On intone, with Answers, pp. 181, 1811. 



TVT. 52.] Multiple and Definite Integrals* 95 



will need to be introduced ; though cents and mils will almost 
certainly come into play. A new relation of value of copper coins 
to our silcer and gold ones will be the whole and the slight altera- 
tion which will be absolutely required. This bugbear of a requisite 
recasting of our copper coinage amuses me ; but what would it be, 
if it were real, to the enormous and really intolerable inconve- 
nience of parting with our shillings and our pounds, or of having 
francs circulating, with only an approximate resemblance to the 
French francs in value, and with too near a likeness to our present 
shillings in their form ? 

* I need not, of course, dear sir, remind you that our present 
word, " Penny," which I by no means wink to get rid of, is used 
now by us, in these countries, with a very different signification 
from the Scriptural one ; and, therefore, why may we not alter its 
meaning a little more ? At all events, I hope that all the foregoing 
remarks may be held by you to have been within the fair limits of 
discussion, and that you will believe me to remain/ &c. 

A Paper by Hamilton, dated Sept. 29, 1857, appeared in the 
Phil. Mag. for the following November. It was entitled: On the 
Calculation of the Numerical Values of a certain class of Multiple and 
"Definite Integrals. The object of the Paper was * to illustrate some 
points in the theory of functions of large numbers, and in that of 
definite and multiple integrals.' In it he extends investigations of 
Poisson, and deals with integers, running to as many as sixteen digits, 
with seven decimals attached ; and speaks of having found the sum 
of the first sixty terms of one integral developed by means of an 
ascending series. Airy, acknowledging the receipt of a copy, 
thanks the author for ' your Paper on tremendous Integrals.' 
Hamilton, however, was not content with the extent of treatment 
given to the subject in this Paper. In the succeeding year he 
carries on its discussion in many letters, two of them of porten- 
tous length, addressed to Professor De Morgan. The first of these 
two letters (commenced Feb. 15th, 1858, and finished May 22nd), 
of which De Morgan writes, that it deserves to be entitled a collec- 
tion of memoirs, extends to seventy-two closely-written folio 
pages; the second (dated July 15th, 1858) to twenty-four similar 



96 Life of Sir William Rowan Hamilton. [1857. 

pages. In the latter he discusses, in comparison with his own 
methods, that contained in a Paper contributed by Professor G. Gr. 
Stokes, Lucasian Professor of Mathematics, and now President of 
the Royal Society, to the Transactions of the Cambridge Philoso- 
phical Society (vol. ix., part i). 

On the 14th of December in this year Hamilton wrote to 
Professor De Morgan a letter of considerable scientific interest: 
its object was, in his own words, to set forth ' a natural mode of 
transition from Argand's method to my own, or from Double to 
Quadruple Algebra ' ' how that passage might hare been made.' 



AETAT. 52.] Festival of Poets. 97 



CHAPTEE XLIII. 

FESTIVAL OF POETS. VISIT TO EDGEWORTHSTOWN. DONATl's 
COMET. PROFESSOR TAIT. BURNS* CENTENARY. 

(1858.) 

IT is to be remembered that, whatever might be the scientific 
investigations to which Hamilton occasionally turned aside, his 
main occupation during the years now arrived at was the prepara- 
tion of his Manual of Quaternions, to which he subsequently gave 
the name of Elements of Quaternions. Becoming aware of the 
imperfection of his Lectures as a treatise, he determined to make 
his second book complete and satisfactory, and to this end he very 
carefully laid down the lines upon which the structure was to be 
raised. He did not anticipate, however, the magnitude which that 
structure would assume, nor the time which its building up would 
occupy. He thought that a volume of 400 pages would suffice for 
a work which 700 did not bring to its completion, and he hoped 
to publish within two years what occupied him to the day of his 
death, more than seven years after the commencement of the 
undertaking. 

It was from this great task that he allowed himself occasion- 
ally such diversions as the Paper on Definite Integrals recorded in 
the last chapter, and the long letters on the same subject which 
employed a portion of his time in the earlier part of 1858. But 
*non semper arcum tendit Apollo.' The reader will be glad to 
catch a glimpse of the laborious mathematician taking a real 
holiday on one fine day of this Spring (the 23rd of April), when a 
few congenial friends visited him at the Observatory. The day 
was frequently referred to afterwards by himself and those who 
VOL. HI. 11 



gS Life of Sir William Rowan Hamilton. [1858. 

enjoyed his hospitality as a festival most happily devoted to 
Friendship and the Muse. For the meeting was one of Poets. 
To Hamilton were added Aubrey De Vere, his dearly loved 
companion- friend ; Mrs. Wilde, the erewhile enthusiastic " Spe- 
ranza" of the Nation ; John Anster, the early translator of 
Faust, a man never to he forgotten hy those who could appreciate 
his fine taste, his wide knowledge, and the summer- lightning of 
his flashing, playful wit; and Denis Florence MacCarthy, an 
accomplished original poet, and known also as a graceful trans- 
lator of some of the plays of Calderon. The well-beloved garden 
and the nearly-equally loved sloping field below the little iron 
gate, were the scenes of merry and serious converse, of recitations 
and readings, including both original effusions and poems by great 
masters of the lyre, recalled to memory in the course of critical 
discussion. A letter of Aubrey De Vere's may be accepted as 
a sufficient reminiscence of the reunion. I insert it in preference 
to a longer and fervid outpouring of MacCarthy's on account of 
its pictorial description of Torquay, the place from which it was 
written. 



From AUBREY DE VERE to SIR "W. K. HAMILTON. 

' ENMOEE, TOEQTTAT, DEVONSHIRE, May 14, 1858. 

' . . . I hope you have been prosperous since I saw you. I 
shall not soon forget the pleasant day we had with you: ourj 
merry dinner, rambles about the green fields, and poetical re- 
citations. I thought we should have met again afterwards,] 
but you never made your appearance in Dublin ; nor was 
more fortunate as regards your friend " Speranza," at whose hoi 
I called in vain. I sent her also the day after our party 
Search after Proserpine, and Mr. De Vere's Misrufr and Mis 
deeds (as some of my English friends call my vindication 
Ireland), but I do not know whether they ever reached her. 
when you meet her next, pray tell her that the books were senl 
the poetical one to abate her appetite for the Waldcuw 
the rebellion* one to revive certain sentiments of patriotism 



AETAT. 52.] Aubrey De Vere. 99 

were once hers, and which would never have been severed from 
JA;/"', had they not been connected too much with certain Repeal 
illusions, and too little with Ireland's true cause what that is 
I will not say. 

' Have you ever been on this southern coast of England? It is 
very lovely bright, beaming, and genial. The villas are so 
thickly scattered, that if the blue of the water were deeper, and 
the green of the fields and trees more harmonised with brown, 
yellow, and that penitential and ashy-grey of the olive, with 
which Italy atones for the luxuriance of her orange groves, one 
might almost imagine oneself on the shores of Salerno. The 
; Ministry seems on its last legs, and the Whigs likely to have 
i their turn again. I suppose the oftener the poets and the critics 
of the political world the Governments and the Oppositions 
have to change places, the better for the sobriety of both parties. 
Adieu.' 

A reference to that day of pleasure made by Hamilton himself 
in a letter to Mrs. Wilde will interest the reader: it tells elo- 
quently how fitted he was to enjoy one of the chief delights of 
spring, and brings out into full view the conflict of feelings which 
his intercourse with his beloved friend Aubrey De Yere then 
stirred up. 



* May \Wi, 1858. ... I am unable to recall so much of 
luman music was there in the poetical party at which you were so 
and as lately to assist whether the birds were singing at that 
ime. This morning I have unlocked the hall-door, that I might 
isten more freely to the storm, the tempest, the whirlwind of 
lelight, and of music, with which the birds are now surrounding 
his house and me. 

'May I pass abruptly to a quite different subject, and express a 
lope that Aubrey De Yere will not succeed in converting, or in 
erverting you? . . . He talked with or to me, for about two 
ours, during our walk to and near Abbotstown, a little before 
our last visit, continuously ; no beauty of Nature seemed able to 
'in him, for even a moment, from his intense contemplation 
f what he regards as the "Glories of Mary"; and I confess 

H -i 



ioo Life of Sir William Rowan Hamilton. [1858. 

that I parted from him with a feeling of fatigue*, though also 
with a sense of admiration, for a high and unselfish which yet I 
must judge to be a mistaken aim. But I do not forget the 
conversation, to me interesting, which I enjoyed with you about a 
year ago, and in which I expressed an admiration of Aubrey, as 
being a " Knight of the Virgin." You know, perhaps, the Lago 
Lugano of Henry Taylor, but will suffer me to copy for you one 
stanza here : 

VII. 

" Our book for us : of amaranthine hues 

The flowers that to the free but searching sight 
Did then disclose their inmost beauty bright ! 

Flowers were they that were planted by the Muse 

In a deep soil which the continual dews 
Of blessing had enriched : no lesser light 
Than what was lit in Sidney's spirit clear, 

Or given to saintly Herbert's to diffuse, 
Now lives in thine, De Vere." 

The lines are not very fine, and on the whole the author of 
Van Artevelde may be said to excel rather in good sense than 
in poetry. But I accept fully his praise of my friend. In one of 
my conversations, at Castle Ashby, with Lady Marian Compton, 
since Alford (who designed that beautiful engraving of the 
Virgin and Passion Flower which you have seen), I said that 
in all my many talks with Aubrey there had been nothing said 
which a lady might not hear.' 

A few brief excerpts from notes to Aubrey De Vere tell of j 
meetings with men by whose conversation he was interested : 

* May 28, 1858. Professor Hennessy, along with Mr. Graham! i 
(the discoverer of the Irish Planet, Metis,) breakfasted 
yesterday, and we enjoyed much intellectual talk, which h; 
scarcely any relation to science, technically so called. Henncssji 

Will you quote against me the lines of Milton (Rook vin.) 

" He ended, or I heard no more, for now 
My earthly by his heavenly overpower'd, 
Which it h;i'l long stood under, strain'd to the height, 

In that rrh Mial collofjuy Mil-liim-, . . . 



AETAT. ,w.] Visitors to the Observatory. 101 

lias since sent me a copy of the Dies irce, dies ilia, of which I had 
only seen a part, namely, the extract in Goethe's Faust. 1 

Hamilton expresses his admiration of this grand old Hymn, 
but his characterisation of it does not seem to me to be sufficiently 
striking or appropriate to call for reproduction. 

' June 10. ... I have been very busy since we met, though I 
have also had a dinner-party here more formal than the one you 
last assisted at. It was, however, as I hope my guests thought, a 
pleasant party. Waller and Anster were two of the company, 
and there were some young men too.' 

A reminder to Anster of the invitation to this party brings 
out some points of character and circumstance not without in- 
terest. 

* June 3. Dr. Waller trill come ; so perhaps you can come 
together ; but if anything induces or enables you to come an hour 
or two earlier, do so. You will find a garden and books, among 
which, I regret to say it, thanks to the partiality of a learned 
lady of several years ago, you will not find a copy of the trans- 
lation of Faust by a certain DOCTOR of my acquaintance. Do you 
know that I think that one of the vainest points about me 
though, that may be saying a great deal is the pleasure which I 
feel in being, as a "Doctor," a fellow of yours and of Waller's? 
I have knelt before two Vice-Chancellors, to receive a Doctor's 
'L'g-ree, and have (must I confess it?) strutted as a "Don" in 
Cambridge, insignificant as I maybe in Dublin society: except 
that you and Mrs. Anster, and a few others, admit me to a certain 
cordiality.' To AUBREY DE VERB. * June 12. Something in- 
terrupted me yesterday, and I have been occupied for several 
hours to-day in receiving a large party of students from the 
College, who came (by appointment) to see the instruments, and 
who have since taken an early dinner with me.' * June 14. 
Another line of mere acknowledgment, to say that I have been 
integrating differential equations at a great rate lately, and that 
\vhcii I hoped to answer some of your notes this morning I was 
ailed on by a very agreeable young American gentleman, Mr. 



IO2 Life of Sir William Rowan Hamilton. [1858. 

Irving Grenell of New York, a nephew of Washington Irving 
(about whom he told me some interesting things), and a connexion 
of some other celebrated people. Another uncle of his fitted out 
chiefly, if not solely, at his own expense, an expedition in search 
of Sir John Franklin. I can't write another line at this moment, 
as Mr. GrenelPs visit has occupied most of my morning : we 
walked together through the fields to Dunsinea, for Miss Rath- 
borne had invited us all to lunch there; but (being busy) I left 
him at her hall-door, though some of my inmates remained there 
to luncheon, and though I had forgotten to eat anything to-day.' 

Hamilton, it will have been remarked, felt himself insigni- 
ficant in Dublin society ; yet, besides the exceptions to which he 
refers, there were eminent members of that society, in various 
degrees of intercourse with him, by whom he was justly appreci- 
ated. Of these was the then Lord Lieutenant, the accomplished 
Earl of Carlisle, who with the following words concludes a note, in 
which he gracefully thanks Hamilton for some presentation copies 
of his recent mathematical and poetical productions : * March 8, 
. . . And now let me me heartily thank you for all the considera- 
tion and kindness I have met with from you.' 

At the end of July, by the invitation of Mrs. Edgeworth,| 
Sir William and his eldest son, William Edwin, joined a la 
party at Edgeworthstown to celebrate a festival in honour 
William Edgeworth, whose father, no longer among the living, 
had been Hamilton's old friend, Francis Beaufort Edgewortl 
and who himself had been schoolfellow of the son who now 
Hamilton's companion. On the return of Lieutenant Edgewoi 
with his regiment from India, the tenants of Edgeworthstown 
combined to testify their joy by the presentation of a sword 
honour, and were, in reciprocal kindness, invited to a banquet 
which congratulatory speeches were made in good old Irish fashi< 
I lamilton enjoyed greatly this opportunity of meeting the mei 
bers of a family with whom he had been long bound in frien< 
ship and Intellectual sympathy. Besides the respected lady 
the head of the family, who was his hostess, he met Dean 



AETAT. 52.] Sonnet on 11 f aria EdgewortJi. 103 

Mrs. Butler, Mrs. F. B. Edgeworth, the widow of his friend and 
mother of the hero of the day and of other children. To one of 
these, who now ranked as Miss Edgeworth, he addressed the 
sonnet which I here present to the reader as the latest extant of 
his verse compositions. A letter to his daughter, written a day or 
two after the festival, tells of his having visited the church. * On 
Friday I went with Eichard to the church tower, and climbed 
(not without danger) to the top of the tower, whereon the spire 
rests, having been erected on a peculiar plan by the late Richard 
Lovell Edgeworth,* father of Miss Edgeworth, the Authoress.' 

The interior of the church contained memorial tablets of the 
Edgeworth family, and among these one met his eyes recording in 
brief simplicity the birth and death of Maria Edgeworth. Under 
the impression of this sight, his thoughts toned down to a corre- 
spondingly stern simplicity, he wrote the following lines : 

To MARY EDGEWORTH. 

Mary ! your great MARIA ne'er from me 

Took tribute of such song as I could give : 

Though deep entreasured in my bosom live 

Thoughts of her frank and loving courtesy, 

Her wisdom, wit, and truth. 'Twas sweet to see, 

Yet sad, to-day, memorial tablets brief, 

Less speaking of the fond survivors' grief, 

Than uttering, in that simplicity, 

Which with her noble nature suited well, 

Such record grave, and not of her alone, 

As fitly might on monumental stone, 

Within those holy walls to which the bell 

From tower ancestral calls with solemn tone, 

Her birth, her death, her name of EDGEWORTH tell. 

' EDGEWORTHSTOWN, July 23rd, 1858. 

I The autumn of this year was saddened to him by the danger- 
ous illness of his daughter, then staying with her cousins at Trim.' 
Her illness was of so alarming a character as to oblige him to give 
up the intention of attending the meeting of the British Associa- 

* See .ZVicAo/son's Journal, vol. ixx. p. 241, December, 1811. 



IO4 Life of Sir William Rowan Hamilton. [1858. 

tion at Leeds, and to summon down to Trim, for repeated consul- 
tation, his friend Dr. Stokes, by whose wise treatment the life so 
precious to her father was, after long struggle and a tedious conva- 
lescence, restored to comparative health. Upon his return to the 
Observatory, in September, the great comet of Donati occupied 
Hamilton's attention as Astronomer, and he incidentally makes 
mention of his being engaged through two entire nights succes- 
sively in observation of the splendid apparition. Much was he 
beset by inquiries on this subject from amateur astronomers, but 
one of these correspondents, the Hon. Mrs. Ward, to whom he had 
been introduced in the previous June by his friend Mrs. Smythe, 
of Farmleigh, proved herself so well-informed, and so much in 
earnest as a student of the science, that he could not but cordially 
assist her design of adding to her record of recent comets some 
history of one which surpassed them all in lustre and magnificence, 
and he obtained for her the grant from the Royal Astronomical 
Society of its Monthly Notices. This accomplished lady, who was 
a cousin of Lord Eosse, and by marriage connected with the 
family of Lord Bangor, published, in 1859, a book entitled, Tele- 
scope Teachings, illustrated by her own drawings, and giving much 
information respecting Donati's Comet. She had previously pub- 
lished a similar work, entitled The World of Wonders, of which 
Hamilton writes, * it might as well have been called Microscope 
Teachings ' ; and also a work on Entomology. Her life was brought 
to an untimely close by a carriage accident in the neighbour- 
hood of Birr Castle, where, as she reports in one of her notes to 
Hamilton, on the gallery of the Great Telescope she 'had more 
than once stood in bitter frost long after midnight.' 

But the month of August brought to Hamilton an introduction 
of a scientific correspondent, who met him on the ground of his 
own special investigations, and was qualified to be in them not 
only a learner but a companion-explorer. A letter from Dr. 
Andrews, Vice-President of the Queen's College, Belfast, sought 
from Hamilton the favour of allowing PETER GUTIIKIE TAIT^ 
then Professor of Mathematics in that College (now the well- 



AKTAT. 53.] Professor Tail. 105 

known Professor of Natural Philosophy in the University of 
Edinburgh), to correspond with him upon the subject of Quater- 
nions. The favour was granted, and it may be said that if it was 
a great advantage to Professor Tait to be instructed and guided by 
its discoverer in this new region of science, Hamilton, also, while 
carrying on his great solitary labour, had reason to feel grateful to 
his able and zealous disciple for valuable sympathy and stimulus. 
The correspondence which began in this autumn was continued 
up to the last year of Hamilton's life with a persistency and 
vigour which must have appeared wonderful to the younger man. 
One incident of it must be recorded. 

Professor Tait had in view to publish, with Hamilton's sanc- 
tion, a set of examples in Quaternions which might afford popular 
proof of the utility of the calculus ; Hamilton, on the other hand, 
desired that its publication might be deferred until his own work 
should appear. So wide was the range, so ample was the detail of 
this work, that its progress was disappointingly slow, and Professor 
T ait's natural eagerness to bring out his own book, of which a 
public announcement had been made, and to turn to other re- 
searches, did at one time cause Hamilton uneasiness, lest the in- 
terest of his Elements of Quaternions might be thus forestalled. But 
this uneasiness was happily dissipated. Hamilton refrained from 
all expression of complaint or remonstrance, and was thus enabled 
to welcome, as it deserved, the honourable spontaneous action of 
Tait, who not only took an early opportunity of publicly acknow- 
ledging, in amplest terms of homage, the supremacy of the dis- 
coverer in the territory of Quaternions, but gave frank engagement 
not to anticipate Hamilton's publication. The cloud passed from 
Hamilton's mind : he was pleased and satisfied with the loyalty 
of his disciple, and in one of the latest notes of his Element** he 
designates him as ' eminently fitted to carry on, happily and use- 
fully, this new branch of mathematical science, and likely to 
become in it ... one of the chief successors of its inventor.' 

* Elements of Quaternions, p. 755. 



io6 Life of Sir William Rowan Hamilton. [1858. 

A very early letter from Professor Tait contained a paragraph 
which gave deep intellectual gratification to Hamilton, both as a 
testimony to the merits of his Calculus, and as an indication of 
the competency of the learner who was beginning to wield the 
instrument. 

From PROFESSOR TAIT to SIR "W. R. HAMILTON. 

1 QUEEN'S COLLEGE, BELFAST, January 3, 1859. 

' . . . About Quaternions in general, I may remark (as indeed 
I very frequently feel), that the processes are sometimes perpkxingly 
easy by which I mean that one is often led in a step or two, and 
without at once knowing it, to the solution of what would be, by 
ordinary methods, a work not so much of difficulty as of labour. 
This, however, I take it, must form one of its great excellences in 
the hands of a person very well acquainted with it. A drawback 
to a beginner, but (as I am gradually being led to perceive) an 
immense advantage to one well skilled in the analysis, is the 
enormous variety of transformations of which even the simplest 
formulae are susceptible : a variety fully justifying a remark of 
yours (Lectures, Art. 504) which, not many months ago, used some- 
times to puzzle me. If I had gained nothing more by reading this 
subject than the facility of making problems and transformations 
for examination papers (especially in Trigonometry), and so saving 
an immense amount of time and trouble, I should have considered 
myself amply rewarded ; but I hope in time to be able to apply it 
to perfectly original work (if anything can be quite original in 
these days). I make these remarks because you expressed your- 
self willing to hear anything I had to say on the subject, and 
because at present they are indissolubly connected with all my 
ideas on Quaternions . . .' 

This letter was seen by Dr. Romney Robinson, who thus com- 
ments upon it : 

From T. EOMNEY ROBINSON, D.D., to SIR W. R. HAMILTON. 

1 OBSEEVATOET, April 18, 1859. 

* I know Tait, and think highly of him. It is very satisfa 
to see one of such energy taking up the Quaternions with a firm 



AETAT. 53.] Professor Tail. 107 

grasp, and from his position he is likely to spread a knowledge of 
their value and power. The surprise which he expresses at the 
startling simplicity of some of the processes must, I think, be felt 
by everyone, and is perhaps that which first opens the mind to 
a full conviction of the extraordinary value of this branch of 
analytic science.' 

It is only due to Professor Tait that I should not withhold 
another passage from a letter written by Hamilton in December, 
1858, on account of the testimony it bears to the progress made 
by Tait in the early stage of his study of Quaternions. 

* ... I have read . . . the first sheet of your " Quaternion 
Proofs," and must say that they appear to me to be wonderfully 
clujanty and to exhibit a very remarkable degree of mastery (so 
far) over the Calculus of Quaternions, used as an instrument of ex- 
]>rt x.s/o/?, and of investigation. It would interest me much to know 
whether (previously to our present correspondence) you had re- 
ceived any assistance from any other student of that Calculus. Or 
did you learn all that you had acquired from the Book itself, com- 
bined no doubt with your own private exercises, of various sorts ? 
If the Lectures on Quaternions have been your only teacher, I must 
consider the result of such a state of things to be not only credit- 
able to your own talents and diligence, but also complimentary to, 
and evidence of, some didactic capabilities of my volume ; which 
ought to tend to console me under my artistic consciousness (as an 
author) of so many faults of execution, that if I could afford the ex- 
pense of bringing out a new edition, I should be more likely to make 
it a new icork. In Dublin, indeed, there exists a little " School " 
of Quaternionists, developed partly by the Lectures and Examina- 
tions which C. Graves and myself have given and held ; and the 
Professor's brother, my old friend, John T. Graves (repeatedly 
mentioned in my Preface), called my attention about a year ago to 
a highly favourable and very elegant article in fob North- American 
/.' '-/Vvr, for July, 1857, on the subject of the Quaternions and of 
my book. But a conscientious author wishes rather to be read 
than to be praised ; and therefore I should like to be informed 
what drew your attention to my book, and whether you had any 
personal assistance in studying it ? ' 



io8 Life of Sir William Rowan Hamilton. [1858. 

In reply, Professor Tait gives an interesting history of his 
study of the subject from the beginning, and concludes 

* So you see that if there is any credit in my progress, it is 
entirely to your lectures and letters that it is due.' 

An early letter in this correspondence contains a short sentence, 
which I must here introduce on account of the light it casts on 
Hamilton's early scientific ambition. In it M. Chasles, the His- 
torian of Mathematics, is called to account for attributing to 
Descartes the discovery of a general property of cones of the 
second order, which had been stated more than 2000 years before 
by the * Great Geometer,' Apollonius Pergaeus. Hamilton's men- 
tion of the name of Descartes prompts the parenthetic addition 
' who has been, however, to me an object of admiration might I 
dare to say of rivalry from my boyhood.' 

In the last month of 1858 Hamilton, in a letter to John 
Nichol, speaks of being ' deeply gratified by a recent compliment 
from Glasgow.' What this was appears from a passage in a letter 
of the succeeding month to Professor Tait, in which he says, * I 
had intended to talk a little about the approaching Centenary 
Festival in honour of Burns, to be held at Glasgow on the 25th of 
the present month, to which I have received a special invitation. 
I consider that invitation to attend "as a guest" upon the occasion 
a highly gratifying compliment, and I wish very much that I 
could avail myself of it, but fear I shall not be able to do so. ... 
I am partly Scotch by my father's side, and always enjoy very 
much my little run to Scotland.' It would appear that, finding 
how deeply Hamilton's feelings had been moved and gratified by 
the public invitation to attend the Festival, his friends Dr. ami 
Mrs. Nichol urged him to make the necessary exertion, and to 
give them for the second time the pleasure of having him for a 
few days the inmate of their Observatory. He found, however, 
that he could not with warrant of conscience yield to the attraction 
of his friends or indulge his admiration for the Scottish Bard. I 
give a letter to John Nichol, written immediately before the day of 



AETA.T. 53.] Robert Burns. 109 

commemoration. It is interesting, as expressing his general esti- 
mate of Eobert Burns, and as revealing links of thought which 
brought the poet specially close to his affection and sympathies. 
Wordsworth's genial and indignant letter in defence of Burns, a 
gift to Hamilton from the author,* had, as the markings in the 
margin show, been read by him with cordial concurrence, and 
he has transcribed at the end of it the fine sonnet of Keats, 
which concludes with the apostrophe : 

' How glorious this affection for the cause 
Of steadfast genius, toiling gallantly ! 
What when a stout unbending champion awes 
Envy and Malice to their native sty ? 
Unnumbered souls breathe out a still applause, 
Proud to behold him in his country's eye.' 



From SIR W. B. HAMILTON to JOHN NICHOL. 

' OBSERVATORY, January 22, 1859. 

* I cannot tell you with what reluctance I bring myself, at the 
last moment, to decline the invitation of your father, and of Mrs. 
Nichol, to be their guest next week during the celebration of the 
Centenary of Burns. 

" 'Twere worth ten years of peaceful life, 
One glance at their array." 

There's a very sincere quotation for you ; but you will probably 
disdain to use it in any speech of yours, as you (most justly) rank 
Burns higher than Scott : I need not say than our Moore, whom 
I may have praised too much (I admit it) at Charlemont House a 
few years ago. But one does not count one's words of praise, in 
speaking of a friend, or even of an acquaintance, recently dead ; 
and I remember that one of my half-sentences (which were well 
received in Dublin] was that " to know Moore, even a little, was to 
love him." 

'Burns was a grand Man. I am not going to praise him; I 

* Vol. i., p. 349. 



no Life of Sir William Rowan Hamilton. [1858. 

leave that to Scotland and to you; supported, and sympathised 
with, by the universal heart of humanity. All this is so very well 
known that it has almost degenerated into common-place. Miss 
Edgeworth once remarked to me that such or such a thing " had 
been said till it was not believed " ! A splendid remark, as I 
thought at the time : but the fame of your Burns can survive it.* 
' I will just tell you one thing, which has pressed itself more 
than once upon my mind in relation to your great National Poet : 
AVhat could have been done for him ? My solution of the problem 
would have been, that he should have been allowed more liberty, 
and have been less interfered with, on the part of his superiors 
in station. In my own case, at all events, I am satisfied that the 
generous (for such to my feelings it has been) non-interference of 
my academical and other superiors, has both allowed and en- 
couraged me to do much more for the public than I was likely 
otherwise to have done. I have, perhaps, purchased Freedom by 
sacrifices ; but, at least, I possess it. As to making him a Prime 
Minister of England, I am of Mr. Burchell's opinion, and in a 
corner cry out, " Fudge ! " Now mind that you will not, the 
least little bit in the world, offend me, if you shall take up this 
remark, abstaining, of course, as you would be sure to do, from 
mentioning me, and tear it all to pieces. How I should like to 
see its fragments fluttering in the midst of some eloquent oration 
of yours ! though I might retain my own opinion after all. 
Affectionately yours.' 



* ' Do you know that I am very strongly reminded of my own father, 
Archibald Hamilton, Esq., of Great (or Old) Dominick-street, Dublin, who died 
while I was very young, and left me only an armorial seal, and a few books, 
chiefly Oriental for he had thoughts of getting me into the East India Sen 
and had some interest in that direction when I look at a clever engraving in 
my possession, of BURNS, receiving the " Wear thou this," from Sc<r 
genius ? ' 



AETAT. 53.] Calvinism. 1 1 1 



CHAPTER XLIV. 

CALVINISM. DATING OF SCIENTIFIC WORK. MANSEI/S BAMPTON 
LECTURES. BRITISH ASSOCIATION AT ABERDEEN. PORTRAIT. 
HERSCHEL ON QUATERNIONS. 

(1859.) 

AN important expression of opinion, on a great religious and 
theological question, is conveyed by Hamilton in a letter written 
early in 1859 to Dr. Mortimer O'Sullivan, a clergyman noted in 
his day for his eloquence in the pulpit, and for ability as a 
controversialist, but less credited than he deserved to be for large- 
ness of view and freedom from personal animosities. This letter 
of Hamilton's is a protest against Calvinism, both as in itself a 
misrepresentation of Christian theology, and as leading, through 
mistaken identification of it with Christianity, to the discredit and 
even rejection of the Christian faith. Although the extract I here 
produce is followed in the original by a request that the letter 
should be considered private, I have thought that I should not 
now be warranted in suppressing Hamilton's opinion on so pro- 
foundly interesting a question an opinion delivered with earnest- 
ness and deliberation by a man who was at once a reverent student 
of Holy Scripture, and a deeply-founded philosopher. 

From SIR W. R. HAMILTON to REV. MORTIMER O'SULLIVAN, D.D. 
1 OBSERVATORY, NEAR DUBLIN, February 25, 1859. 

* In the midst of mathematical and other engagements, I feel 
an irresistible impulse to thank you for the publication (not the 
gift) of a couple of anti-Calvinistic sermons which have fallen into 
my hands within the last day or two. 



112 Life of Sir William Rowan Hamilton. [1859. 

* To me it appears to be of vital importance, that a protest 
against Calvinism should be raised by people like you if there be 
many such. And this I say, having a very dear sister living, 
whom I very much respect as well as love, and who is a devoted 
Calvinist ; being also notwithstanding, as I might almost be 
tempted to say but really I ought not to say it an extremely 
pious and practical Christian. 

* With her deceased friend, Mr. Krause, I had, through her, 
some very slight acquaintance, which I valued. The beauty of 
holiness was seen in the man, and appears also in his sermons, of 
all of which I have purchased copies. But even Mr. Krause has 
completely failed to turn me into a Calvinist. 

* I repeat that I regard this question as a vital one as of course 
the Calvinists themselves would say. If the alternative were pro- 
posed to me, Whether I would be a Calvinist or an Infidel ? I 
dare not finish the sentence. 

4 But that unwritten conclusion has been, is, and will be 
thought, spoken, written, and acted out, by other men. I have 
studied, to some extent, the infidelity of Burns, of Byron, and of 
Shelley ; and am satisfied that it was not Christianity, but Calvin- 
ism, which revolted them. May God have mercy on their souls ! 
and on those who misrepresented Christianity. 

* Of course I have read remarks on the " happy inconsistency " 
of human nature, and all that. Very consoling, to be sure. But 
still I shall be happy to be informed that you proceed to denounce, 
and batter, with a sledge-hammer power of reasoning, and also with 
all that wit, grace, fancy, and (if I dared to say it), humour, which 
you cannot help admitting into your publications, a System of 
Theology, which, if it had been imposed by external authority 
upon me, would have gone near to making me an infidel.' . . . 



The value set by Hamilton on truth and accuracy, prized by 
him in all matters, but here inculcated in reference to the veracity 
of scientific publications, is proved by the following letter to the 
present Provost of Trinity College, who at the time was one of 
the Honorary Secretaries of the Royal Irish Academy. Hamilton's 
insistence upon the obligation of exactness as to dates, &c., in pub- 
lications affecting, it might be, questions of priority in dis- 






\ in \r. 53.] Accuracy in Dating. 113 

covery, could not have been addressed to anyone more entirely 
in sympathy with him as a lover of truth and justice than 
Mr. Jellett. 

From SIR W. E. HAMILTON to KEY. J. H. JELLETT, M.A., F.T.C.D. 

[FROM A COPY.] 

' OBSERVATORY, March 27, 1859. 

*I have received, by this morning's post, a small packet from 
Gill containing a proof and duplicate of my last communication 
to the E,. I. A. I suppose that it is you whom I have to thank for 
this prompt attention to my wishes. In the cover, Mr. Dillon, for 
Mr. Gill, requests me to return the proof on Monday, that is to- 
morrow ; for on this occasion at least I am compelled to borrow 
the Sunday for work. 

1 As you know that I have stood up for the rights of authors, 
or rather for one form of these rights another of them con- 
sisting in a moderately rapid publication of their Papers you will, 
perhaps, permit me to say that I also respect very highly, and 
would, if it were needful, endeavour to maintain the Prerogative 
(or more accurately) the Duty of an Editor. 

' In the present case I admit that all requisite courtesy to the 
Author (meaning myself for the moment, as people usually do 
mean themselves, when they talk in an abstract style) is dis- 
charged as in this last case when one clear day is allowed 
(although, if it could be made a week-day on some future occa- 
sion, I should be better pleased) for the opportunity of that 
author's revision. 

'And by revision I do not mean re-writing. Society allows 
some mystification ; but the more absolutely pure, and sacred, 
scientific inter- communication can be kept from even ha //-truths, 
in statements, the better shall I be pleased ; and (what is much 
more important) the better will the scientific world get on. 

4 Accordingly, I have made only the most minute, trifling, and 
purely verbal alterations hardly five words being inserted or 
changed in the proof which has this day reached me. I confess 
that I desire though it is one of the slightest things that I wish 
for to take a date when I communicate a Paper. 

* Now, if authors are to be permitted to add and alter, beyond 

VOL. III. I 



H4 Life of Sir William Rowan Hamilton. [1859. 

some very narrow and fixed limits, I do not see how they can take 
any such date, in virtue of a printed record, so liable to be evaded. 
In my own case, I am resolved that there shall be no such evasion. 
Nor is this any new resolve of mine. If you will take the trouble 
of turning over those parts of the volumes of the Transactions of 
the E. I. A. which contain Papers of mine, you will see that 
I have on several occasions taken pains to state that I had availed 
myself of permitted opportunities to recast whole memoirs. 

'I wish to annex a note but under the head of Note added 
during printing the insertion of which will cause (as I count) 
scarcely any trouble or delay. Tell me something about the 
Exiles, when you have a minute, and believe me/ &c. 

A note of earlier date furnishes an explanation as to the 
'Exiles': I give it because it shows that Hamilton, though not 
having been much in intercourse with Mr. Jellett, was attracted 
by his powers and character. It incidentally reveals also other 
facts not without interest. 

From the SAME to the SAME. 

' OBSERVATOKY, March 17, 1859. 

* Our little discussion at, or rather after, the Meeting of the 
Academy, which I must confess that I enjoyed, as I do everything 
which leads me to know more of you, will not (I hope) have 
disinclined you to take charge of the Two Pounds enclosed : one 
for the Groldsmith statue or monument, to which I have from the 
first designed to contribute ; and the other for the Neapolitan 
Exiles, with whom I see your name associated as a receiver of 
subscriptions. 

* P. S. I often make it a condition that my name shall not be 
published, but on the present occasion it appears to me that a part 
of the value of the subscription will depend on the publication of 
the name.' 

The Bampton Lectures of Dr. Mansel, which provoked against 
them the impassioned reclamation of Frederick Denison Maurice, 
were at this time the subject of discussion in Dublin as elsewhere, 



AETAT. 53.] Mansers Bampton Lectures. \ \ 5 

and Hamilton was referred to for his opinion of them by his old 
frit -ml, Dean Butler of Trim, writing on behalf of his neighbour 
the Rev. John C. Macdonnell, then incumbent of Laracor.* 

The following letter addressed to Dr. Lee, Fellow of Trinity 
College, Dublin, and well known as a Theologian, exhibits a 
partial result of Hamilton's perusal of the book. It is to be 
wished that it met more fully the main contention of the author 
as to the nature of our knowledge of God, but what is said in it 
has a value that will be recognised : 

From SIR W. B. HAMILTON to EEV. WM. LEE, D.D., F.T.C.D. 

' OBSERVATORY, July 2, 1859. 

'I am much obliged by your kind note received this morning 
on the subject of Hansel's "Bampton Lectures"; and, if it be 
possible, am still more gratified by your having been so good as to 
remember our little conversation respecting them at Jellett's party 
lately. 

* I suppose that I told you that my valued friend, Dean Butler 
of Trim my acquaintance with whom wants only a few months 
of being one of forty years ordered not long ago his Dublin 
bookseller to send to me his copy of those Lectures for my 
perusal; with a request that I would give him my opinion 
upon them, especially as bearing on some points of philosophy, 
with which he was pleased to suppose that I might be familiar. 

1 Accordingly, I received the book, and have it by me still. 
One careful and continuous perusal I have already given to the 
Lectures themselves besides some dipping into the Notes with 
an ungrudging and unbounded admiration. Such is, I think, the 
proper temper with which to approach a work of genius and of 
learning, devoted to the highest interests of religion and huma- 
nity. But after one such admiring, and indeed reverential, pupil- 
ine, of sitting at the feet of a new author, a student may be par- 
doned for presuming, or rather an honest teacher would (no doubt) 



* After wards Dean of Cashel, now Prebendary of Peterborough : author of 
Donne/Ian Lectures on the Doctrine of the Atonement. London, 1858. 

I 2 



ii6 Life of Sir William Rowan Hamilton. [1859. 

desire him, to take for awhile the converse attitude, of judge and 
critic. You remember, probably, that passage of the Kritik* in 
which Kant speaks of the necessity of Man's assuming that 
attitude with respect to Nature itself: "An die Natur gehen, 
zwar um von ihr belehrt zu werden, aber nicht in der Qualitat 
eines Schiilers, der sich Alles vorsagen lasst was der Lehrer will, 
sondern eines bestalten Eichters, der die Zeugen nothigt auf die 
Fragen zu antworten die er ihnen vorlegt." 

6 Accordingly I am, just now, in a mood for finding fault with 
Mansel trying, as it were, to pick (in thought) a quarrel with 
him : I hope not in revenge for his having extorted from me so 
large a share of admiration. 

* The grand question which he raises in my mind, at this 
moment, is the following : 

'Does he insist too much on the limitation of the human 
faculties ? 

' For example, " Philosophy " is a grand and sacred word. 

" How charming is divine Philosophy ! 
Not harsh, nor crabbed, as dull fools suppose, 
But musical as is Apollo's lute." 

4 1 therefore do not presume to assert nor to deny that there 
may exist a " PHILOSOPHY of the Infinite" But I am sure that 
there does exist a "SCIENCE of the Infinite" -, to wit, Mathematical 
Science. And with respect to Note 15 to MansePs Lecture II. 
to which Note Dean Butler, on behalf of a friend of his and mine, 
has recently invited my attention I must just be permitted to 
repeat from it, though not with the author's application, four 
pregnant words : " Callide, acute, nihil supra." 

6 This, however, is far from being my judgment, even in its 
second and (transitionally) hostile stage, respecting the lectures at 
large. I shall not be contented till I have read them several times 
over. 

* Just now it seems to me that if there be anything whidi I 
shall have a difficulty in finally adopting, it will be the 8ev< 
and almost contemptuous, criticism of Dr. Mansel upon Km it's 
Practical Reason (Kritik der Practise/ten Venwnft) : although 
perhaps more hinted than expressed. 

4 1 have possibly been bribed to like that work of Kant, by tlu 



AETAT. 53.] Meeting at Abe yd ecu. 1 1 7 

circumstance of its having been mentioned to me in conversation 
for our intercourse was not always monologue by my illustrious 
friend, and (if I may dare to say so) Master, Samuel Taylor 
Coleridge ; who gave me his German copy of the Urtheikkrqft, 
through his own particular ally, Joseph Henry Green. 

' May I confess that I have only as yet heard of your own work 
on the Inspiration of Holy Scripture ; and will it be presuming 
too much, if I venture to suggest an exchange of publications ? 

' At all events, I am about to write your name in a copy of my 
Lectures on Quaternion* : a very interesting and valuable commen- 
tary on a part of which work has been quite recently sent to me 
from Leipzig, by the writer of that commentary or article, namely, 
by Professor Moebius, author of the Barycentric Calculus, &c. 

' And if you are so obliging as to send for me to the Royal 
Irish Academy a copy of your "Inspiration," I shall not only be 
personally gratified, but will undertake to read the book. I am,' &c. 

Dr. Lee, in acknowledgment of this letter, states his view of 
the subject in words which may be read with interest. He says: 

' I have re-read Mansel during the summer with great pleasure. 
I am sure that the popular misrepresentations of his view arise 
from not noticing the broad sense in which he employs the term 
Theology, viz. the Science of the Divine Nature. Hegel and others 
regard this science as within the reach of human reason ; and in 
opposition to this exaggerated estimate of man's faculties Mansel 
protests. I think I mentioned this to you already, and a re-perusal 
of the Lectures confirms me in this opinion. He may, of course, 
following your namesake's theory, err in his notion that the 
Infinite can in no sense be an object of Science; but this is a 
collateral question.' 

At the end of July Hamilton escorted his daughter, still an 
invalid, to stay for some time with the Keating family, relations 
on her mother's side, at Ehyl, in North Wales. In the beginning 
of September he carried her off from these kind friends to become 
the guest of his own maternal relations, Mr. and Mrs. Joseph 
"Willey, at Fulneck, the Moravian settlement near Leeds. Here 



1 1 8 Life of Sir William Rowan Hamilton. [1859. 

she was to remain while he attended the Meeting of the British 
Association at Aberdeen. It was presided over by the Prince 
Consort (whose opening address he speaks of having 'greatly 
enjoyed'), and brought together many scientific friends whom 
Hamilton was glad to meet again. AmoDg the most eminent 
of these were Faraday, Lord Eosse, Airy, Brewster, Eomney 
Robinson, Humphrey Lloyd, Murchison, Sabine, Henry Smith, 
the Abbe Moigno. But perhaps the personal contact most inte- 
resting to him on this occasion was his meeting in the flesh his 
epistolary friend, Professor Tait, by whom was introduced 
to him a man who, as we have seen,* became, after Hamilton's 
death, a successful employer in physical researches of the Calculus 
of Quaternions, the gifted James Clerk Maxwell. Hamilton's 
own contribution to the proceedings of the Association was a 
Paper On an Application of Quaternions to the Geometry of kernel's 
Wave- Surf ace. A special interest was attached in the mind of 
Hamilton to this communication, because, by an agreement be- 
tween him and Professor Tait, both had been carrying on simul- 
taneously, but in entire independence, an investigation by 
Quaternions of the same problems. Hamilton distributed a 
lithographed statement of his method of solution, besides making 
it the subject of an oral address to Section A, in preparation for 
which he pencilled in a manuscript book a note, which I here 
insert as showing what was his aim at this time in the develop- 
ment of his calculus : 

' Not about to lecture on Quaternions ; nor do I pretend on 
the present occasion to produce any new property of that celebrated 
and important surface, which is mentioned in the title of this com- 
munication, but merely to exemplify the conciseness and simplicity 
with which the language of quaternions enables those who have 
acquired. some familiarity with that language to express and com- 
h'mi' general conclusions, and even physical hypotheses, and then to 
Innntform the resulting formula), and to interpret the new equations 

* See vol. ii., p. 445. 



AKTAT. 53.] Portraits of Hamilton. 1 19 

BO obtained. I said hypotheses : for I desire it to be distinctly 
understood that I do not by any means adopt any such hypotheses, 
nor at all pretend to express, or even to form, any opinion of my 
own on such a question as this : Are the vibrations of the ether (if 
such vibrations actually exist) perpendicular or parallel to the plane 
of polarisation ?' 

At Aberdeen Hamilton was the guest of Mr. James Westland, 
and, as on former occasions of a like kind, he appears to have 
gained the esteem of the whole family circle of his host. He 
enjoyed also, with a distinguished party, the hospitality of Mr. 
and Mrs. Alexander Thompson, of Banchory House, and was one 
of the two hundred members of the Association who, on the invi- 
tation of the Queen and the Prince Consort, visited Balmoral a 
visit which untoward circumstances rendered less successful than 
had been hoped. Leaving Aberdeen for the south on the 23rd 
of September, he received the saddening intelligence of the death, 
on the 19th, of his much loved, accomplished friend, Dr. J. P. 
Nichol, of the Observatory of Glasgow. The friendship between 
them had been of quite a brotherly character, scientific know- 
ledge and pursuits being in their case the acquisitions and employ- 
ment of natures which derived their most prized enjoyments from 
elevated sentiment and poetic thought. This loss had the effect of 
binding Hamilton only the more closely to the son and daughter 
of his friend. 

To rejoin his own daughter at Fulneck was now his aim, and at 
this Institution he again spent some days, the principal incidents 
of which were his giving a Lecture on Astronomy to the youthful 
pupils of both sexes (from whom letters of thanks were addressed 
to him*), and by his sitting, with his daughter standing at his 
side, for the photographic likeness from which an autotype repro- 
duction has been placed as a frontispiece to the second volume of 



* I have found a similar letter of thanks from the girls of Mercer's School, 
situated in his own parish of Castleknock. 



I2O Life of Sir William Rowan Hamilton. [1859. 

this work. And here I take the opportunity of expressing my 
opinion that this representation of his features stands out from all 
other photographs of him which I have seen (and I believe I have 
seen almost all that were taken), as alone doing something like 
justice to the combined intellectual and moral character of the 
subject. It exhibits, I think, both in conformation and expression, 
the profound thinker, the reverent benevolent sage. The marble 
bust in the Library of Trinity College is from the hand of Foley, 
and a photograph from it supplies the frontispiece to the present 
volume. Our eminent sculptor never had the advantage of seeing 
Sir W. B. Hamilton, and had to work from small photographs 
and a cast of the anterior half of the head. The aspect which the 
photograph presents will, however, be acknowledged by all who 
knew the living man to be both fine and like.* Hamilton had 
now the happiness of recognising that the visits in England of his 
beloved child had been attended by the wished-for result, and in 
restored health she accompanied him on his return to the Observa- 
tory at Dunsink. 

"When again at home and at work upon his new book, he 
received from Sir John Herschel, to whom he had sent a copy of 
the Aberdeen lithograph of his quaternionic treatment of Fresnel's 
"Wave, the following * cry of distress.' 



* In the course of this year Hamilton was much, and, as it appears to me, 
rather unreasonably disturbed by a suggestion made by General Larcom, that 
its ex-presidents should consent to their portraits being obtained for the meet- 
ing-room of the Royal Irish Academy. He felt that he ' could not afford ' to 
present his own, and shrunk from saying so, and he strongly objected to what 
Roumey Robinson, who sympathised with him, called * the clumsy machinery 
of subscription' being resorted to, at least during his lifetime; this he con- 
sidered would be ' humiliating ' to him. He seems not to have considered that 
a good portrait could only be taken from life. The Academy and his country 
have, I think, great reason to regret a decision which conduced to the in ill- 
existence of a si-rii-s of portraits which would have been of permanent historical 
interest. I refrain from printing the correspondence on the subjrct. with 
Dr. Robinson and General Larcom, who was astonished at the ill reception 
which his proposal met with. 



I-;TAT. 54.] Her schef s Cry of Distress. 121 



From SIR J. F. W. HERSCHEL to SIR W. R. HAMILTON. 

'37, TAVISTOCK-PLACE, RITSSELL-SQUABE, November 18, 1859. 

4 MY DEAR SIR WILLIAM HAMILTON Your deduction from 
Quaternions of Fresnel's "Wave is one of those things which I 
have just knowledge enough to admire without enough to under- 
stand. But it set me again on reading your Lectures on Quater- 
nions, and I got through the three first chapters of it with a much 
clearer perception of meaning than when I attacked it some three 
or four years back, but I was again obliged to give it up in 
despair. Now I pray you to listen to this cry of distress. I feel 
certain that if you pleased you could put the whole matter in as 
clriir a light as would make the Calculus itself accessible as an 
instrument to readers even of less "penetrating power" than 
myself, who, having once mastered the algorithm and the conven- 
tions so as to work with it, would then be better prepared to go 
along with you in your metaphysical explanations. 

* Do pray think of this. At the risk of offending, I will venture 
to say you will not have done yourself justice if you do not give 
the world some clue that a lower class of thinkers can unravel than 
those who alone can hope to master that book. 

* The simplest way would be to give forth a number of examples 
of the treatment of problems and theorems by it. I mean not 
examples which shall be of themselves general theorems or impor- 
tant discoveries, but good honest ordinary problems or theorems, 
such as can be readily worked by common Algebra and Trigo- 
nometry, but gradually increasing in difficulty ; and these might 
be prefaced by a clear statement of the Rules of the Calculus as 
Rules. 

'Such a book would have an immense influence. Hundreds 
would learn to use the Calculus as a means of investigation and 
its theory would by degrees [be] popularised. Pray excuse this 
from yours very sincerely.' 

Hamilton replied in grateful terms, and enclosed for Herschel's 
satisfaction the initial sheet of his proposed new work, written, as 
he trusted, ' in a style like that which you desire for me ; or at 
least more like it than the Lectures.'' He had the gratification of 



122 Life of Sir William Rowan Hamilton. [1859. 

receiving from Herschel the following acknowledgment of his 
specimen : 

From the SAME to the SAME. 

1 November 25, 1859. 

' Nothing can possibly be clearer or more to the purpose, and 
if the rest of your book be as much so, and go on the same prin- 
ciple, it will be one of the most important, and I will venture to- 
say, the most widely circulated elementary work ever published/ 

A second sheet was sent by Hamilton to Herschel, and drew 
from him the criticism that by the introduction of some difficult 
applications of the Calculus the attention of the commencing 
student was too soon diverted from its principles. Hamilton 
submitted to the criticism, and re-wrote the sheet, thanking his 
critic by sending to Lady Herschel for communication to him the 
passage from Horace's Epistola ad Pisones (1. 438. . . .), in which 
the Roman teacher of his Poetic Art exhibits in Quintilius Varus 
the character and action of the critic who was a real friend. In a 
subsequent letter of Herschel, acknowledging further proofs, he 
first suggests the suppression of a passage respecting transversals, 
and then retracts the suggestion, and concludes with the words, 
'Au reste the thing is CHARMING, and I can only add, Gro on 
and prosper/ 



AETAT. 54.] Anharmonic Co-ordinates. 123. 



CHAPTER XLV. 

ANHARMONIC CO-ORDINATES. GEOMETRICAL NETS IN SPACE. GKO- 
METERS IN TRINITY COLLEGE. CIRCLE-SQUARING. AUBRKY 
DE VERB'S 'INISFAIL.' 

(i860.) 

IN the beginning of 1860 Hamilton, while carrying on his work 
upon the Elements, opens a mathematical correspondence with 
Dr. Hart, then Senior Fellow of Trinity College, now Sir Andrew 
Searle Hart, Yice-Provost. From problems connected with a cir- 
cumscribed pyramid, and tetrahedra in general, he rapidly advances 
to what may be called a new calculus, independent of Quaternions, 
though advantageously employed in connexion with them, the 
calculus, or instrumental machinery, of Anharmonic Co-ordinates ; 
and on the 27th of February he begins what he himself calls a ' pro- 
digious' letter, which, becoming an extended treatise, reaches at 
last the 216th folio page of closely-written work, and is completed 
by a postcript of sixty-four similar pages, dated the following 28th 
of August. Some of the results arrived at were, during its pro- 
gress, communicated to Dr. Salmon, touching, as they did, some of 
the problems published by him in his Higher Plane Curves, and he 
acknowledges their novelty and importance, while Dr. Hart in a 
letter, dated so early in the correspondence as April 6, says of 
them: 

'If your Anharmonic Co-ordinates should never be used again, 
they have already conferred a great benefit on geometry, by turn- 
ing your attention to the large and much-neglected field of cubics, 
in which you have already filled many gaps, and placed in a new 
point of view a theory which we have been hitherto studying 
entirely through Salmon's glasses. If you succeed in compressing 



124 Life of Sir William Rowan Hamilton. [i860. 

so much matter into the modest limits of an " Appendix," it will 
be a wonderful achievement in the art of condensation.' * 

In letters written to Aubrey De Vere on the 16th and 17th of 
March, Hamilton gives to his friend an interesting aperfit of 
what he had been doing on this subject. 

* March 16. Within the last three weeks I have written quite 
an essay not to say a treatise on (what seems to me) a new 
scientific system, or method, of what I call " Anharmonic Co-ordi- 
nates " ; and which, wonderful to tell, appears new, also, to all the 
geometrical friends whom I have consulted in our University, e. g. \ 
Graves, Salmon, Ingram, Hart. It is strange that after a couple j 
of millennia, a thought, which seems to be traceable to Euclid 
(through Pappus) should be found now to admit of a vast and 
unforeseen expansion. . . .' 

Forwarding a copy of his sister's poem on Columbus, which his 
friend had asked for a month previously, he adds : 

' March 17. ... One would think it was some heroic effort of 
virtue, my writing out a few lines of poetry for a friend : I put it 
off so long, and seem to make so much of work or fuss about it ; 
although when I do begin, I experience really great pleasure in 
the act. If you could only see what quantities of less interesting 
(viz. mathematical) matter I write to my scientific correspondent s ! 

* Such an Appendix is not to be found in connexion with the Element* <>/ 
Quaternions, into the early part of which work are, however, introduced Sec- 
tions dealing compendiously with both Anharmonic Co-ordinates and Geometri- 
cal Nets in Space. But from the fact that at the foot of page 34 reference u 
made to * Note A on Anharmonic Co-ordinates,' and at the foot of pages 35 
and 56 to 'Note B upon the Barycentric Calculus' (' Nets in Space '), which 
Notes are non-existent, it would appear that the author intended to annex an 
Appendix giving further details as to both these subjects. It must be reineni- 

<1 that the Elements of Quaternions was published after the death of the 
author, and is in an incomplete state. On the subject of Anharmonic Co-ordi- 
nates he made a communication to the Royal Irish Academy, which is printed 
in the Fi-uwdiniis I'm- the Session of 1859-60 ; and on the 24th of June, 1 

' was read by him, before the same Institution, a very important Paper, 
extending in print to above 50 pages, on Geometrical Nets in Spuce, u I \ijiei 
carrying on to 'a Quinary Calculus for Space,' the researches \\hieh hud in 
tin ir earlier stage furnished him with his method of Anharmonic Co-ordin 



\i-T\T.54.] Geometry in Trinity College. 125 

much faster, indeed, including the composition, than my assistant, 
or my son, can copy them for me. But I suppose that a man has 
instinct, as well as reason ; the aropyri of the animal creation exists, 
and shows itself with me, when a new conception in science has 
dawned upon my intellect, and is in danger of altogether perishing, 
if not duly incubated in its season. My last conception, of a 
"I'frical kind, has taken my friends in Dublin who, on such 
subjects, are admitted to hold a very high place in the scientific 
world entirely by surprise . . . But I expect to be quite cool 
again upon the subject in about a month. You once confessed to 
me that for a day or two after writing a new sonnet, you were not 
an impartial judge of it.' 

I insert here extracts from the long letter to Dr. Hart, which 
have an interest derived from their recognition of more extensive 
knowledge than his own in modern geometry on the part of his 
mathematical colleagues in Trinity College. They are pleasing 
testimonies of his un jealous disposition, of his habitual desire 
to acknowledge points of superiority in others. 

C. 1860, p. 38.* February 27, 1860. I remember staring ten 
or twelve years ago when Townsend told me [in 1847], in what I 
call Brinkley's Garden, that a plane conic has double contact with 
each of its foci considered, of course, as infinitesimal circles ; and 
he admitted that *,/, k, which were then comparatively recent, had 
nothing in them more paradoxical, on their first appearance. You 
see that I have made a step or two forward since then, aided (no 
doubt) mainly by Salmon's books though I know something of 
foreign ones also in what is called the " Modern Geometry " ; 
but I have the deepest feeling of my inferiority, in that respect, 
to persons who have made the subject their special study.' 
P. 151. 'April 25. As I cannot expect that you will do more 
than skim, at present, the sheets of this prodigious letter ... I 
isolate thus a question or two in the hope of your answering them 
separately. You have been very useful to me already by calling 
my attention to parts of Salmon's book, which I had not sufficiently 
considered. His "Lessons," &c., I have got, and admire.' P. 241. 
' The parallelism of ff'll to A, C / in figures 29 and 30 did not 
occur to me when I saw these figures, and 'tis perhaps curious that 



126 Life of Sir William Rowan Hamilton. [i860. 

I saw it half an hour ago by reasonings about conic* many minutes 
before I was able to verify it by any special argument from 
circles. Such verification, however, should certainly have offered 
itself sooner to any of your Junior Freshman friends. (I was of 
course a Junior Freshman in my time, and was never beaten in 
Geometry nor, as it chanced, in any other subject in my divi- 
sion ; but things have so much advanced generally in our Univer- 
sity since then, that I might well meet with a different fate, in 
every department, if I could go in again now, as I have sometimes 
half -wished to do, to be lectured by Salmon, and Townsend, and 
others.) . . .' 

This year brought to Hamilton the commencement of a corre- 
spondence, very unwillingly entered upon, with Mr. James Smith, 
of Seaforth, Liverpool, who about this time actively challenged 
notoriety as a squarer of the circle.* Hamilton felt obliged to 
repress his pretensions with greater severity of tone than was usual 
with him ; I should scarcely have referred to the incident, had not 
the result been that in the course of the discussion he was led to 
communicate to the AthencBum (June 8, 1861), and afterwards in 
an extended form to the Philosophical Magazine (April, 1862) an 
4 Elementary Proof that Eight Perimeters of the Regular Inscribed 
Polygon of Twenty Sides exceed Twenty five Diameters.' Archi- 
medes had proved more than 200 years B. c. that 71 perimeters 
of a regular polygon of 96 sides inscribed in a circle exceed 223 
diameters, whence easily followed the theorem that eight circum- 
ferences of a circle exceed twenty-five diameters : a theorem 
contradictory to Mr. James Smith's assertion that they are exactly 
equal. Hamilton's * Elementary Proof ' served as an d fortiori 
argument to the same effect. 

The death of Hamilton's youngest sister, Archianna, occurred 
in February of this year, when she was staying with her sister 
Sydney in Dublin. The event called into active exercise his 
affection and sympathy towards both his sisters. After an inter- 



*8ee De Morgan's Budget of Paradoxes, p. 318. . . . , and 



IETAT. 54.] Deatli of Arcliiainm Hamilton. 127 

val it was followed, as had been the death of his sister Eliza, by a 
visit to the county of Wicklow. In that well-known region's diver- 
sified beauties he sought as before, and experienced, the restorative 
influences of nature. 

Extracts from Hamilton's correspondence with Aubrey De 
Vere principally on literary topics may fitly close the record of 
this year, and lead on to that of the next. 

From AUBREY DE VERB to SIR W. R. HAMILTON. 

'CuBBAGH CHASE, February 21, 1860. 

1 . . . It would be well worth your while to look at a translation 
by D. F. MacCarthy of one of Calderon's " Autos Sacramen tales " 
which appeared in the number for July, 1859, of the Atlantis. It 
is a very wonderful sort of poetry which has become better known 
in Germany than in England . . . .' March 15, 1860. ' I must 
just send you a line to tell you how sorry I was to hear of the 
death of your sister. I remember her perfectly. As life goes on, 
how many of those connected with one's earlier and brighter 
recollections have taken their place with those who have gone 

" From sunshine to the sunless land." 

4 ... Alas for the good old times, when we used to talk of 
Francis Edgeworth ! ' 

From SIR W. E. HAMILTON to AUBREY DE VERE. 

' November 30, 1860. 

' I received your " Select Specimens of the English Poets," 
this morning, with very great pleasure, and have read much of it 
already, with the enjoyment for the larger part of recognition : 
but also with the new pleasure of reading your thoughtful, tempe- 
rate, and wise remarks on the various authors, or rather (for 
the present purpose) poets, discussed. I also received about a 
fortnight ago a copy of your father's " Julian," and " The Duke 
of Mercia," which I value very much; and not the less, because 
I have certainly among my Curragh books, or memorials in- 



128 Life of Sir William Rowan Hamilton. [i86(h 

eluding presents from your father a copy of at least "The Duke- 
of Mercia." But "Julian the Apostate" had to me the air 
of a quite unknown, and of a wonderfully beautiful composi- 
tion/ 

From AUBREY DE VERB to SIR W. E. HAMILTON. 

' LONDON, July 19, 1861. 

* I have requested M'Glashan and Grill to send you a new 
volume of poetry which I have just published. . . . The most 
important poem in it is one called " Inisfail," which occupies the 
latter half of the volume. When you have time to look into the 
book, pray begin with this poem, for I would like to have your 
opinion of it. It is a somewhat new idea in poetry, I believe. It is 
a Chronicle Poem, intended to represent in their essential features, 
both the history and the inner life of a nation and race, and to do 
so chiefly through songs and ballads, such as might have proceeded 
from the bards contemporary with the events described. There 
never was a more wonderful history than that of Ireland. Except 
a few picturesque and pathetic details, people commonly know 
nothing about it. They do not suspect that, seen from one point 
of view, and that the highest, so far from being a jumble of sad 
fragments, it has the highest possible Unity and Significance. 
Doubtless my point of view may not be that of my readers, in 
some cases. But poetry has a right to its Postulates, has it not ? 
I cannot judge of the execution of course ; but the theme is a 
good one, I think. Nations are the long-lived Patriarchs still 
allowed to survive, and work out their Destinies, on earth. They 
have a continuity not now given to the life of an individual ; and 
their fortunes have thus an affinity with song.' London, Nov. 4. 
' A few of the Reviews at this side of the water have been 
snarling at the Irish tone of my Chronicle Poem of " Inisfail," 
though I meddle with nothing within the last century. They 
cannot afford, apparently, even an epitaph to the Ireland of six 
centuries! What have you been about of late? Writing on 
mathematical subjects, I suppose, as usual. But you must find 
time too for an occasional sonnet also "golden mile- stones " 
up along the road of life. You were sorry, I am sure, for Eli/a- 
beth Barrett Browning.' 



AKTAT. 54.] Recognitions from Abroad. 129 



CHAPTER XLVI. 

RECOGNITIONS FROM ABROAD. HONORARY DEGREE AT CAMBRIDGE. 
TAIT. INGLEBY. KIRKMAN. 

(1861.) 

THE beginning of 1861 brought Hamilton from abroad gratifying 
proofs of the extending study of Quaternions. Returning to Dr. 
Waller, Editor of the Imperial Dictionary of Biography, a proof, 
sent for critical correction, of a notice of his own life and works, 
he thus writes : 

i April 9, 1861. ... I do not wish anything additional 
whatever inserted. ... I think it may interest you to know 
that Dean Graves had lately received for me a copy of a very 
interesting and valuable work, in Italian, on the Quaternions, 
(Calcolo dei Quaternioni di W. R. Hamilton, &c.) by Professor 
Bellavitis ; addressed on the outside, Al Chiarissimo Signore Prof. 
W. R. Hamilton, &c., which address my daughter (don't expose 
her) read out as if it had been Al Carissimo (" Fame is love 
disguised." Shelley). But this is only the last of several recog- 
nitions and more than mere recognitions, since they attest 
successful study of the Quaternions, which I have received from 
foreign countries, especially from Germany and America ; for 
France is very slow to accept any truly new idea, although she 
works it well, when she has once caught hold of it. Something 
like a century had passed before Newton was appreciated in 
France ; but the world moves now at a somewhat faster pace. I 
have long been, however, one of the six mathematical correspon- 
dents technically so called of the French Institute. (At least, 
I think the number is six ; and am sure that it does not exceed 

;/, out of the whole world which to France is " barbarous," in 
the old Greek sense) . . . ' 

VOL. ITT. K 



130 Life of Sir William Rowan Hamilton. [1861. 

Recognition of Hamilton's work on a different subject from 
Quaternions, namely, Systems of Rays, arrived in February from 
Berlin. Dr. Meibauer presenting to the Eoyal Irish Academy 
his Paper entitled De generalibus et infinite tenuibus luminis fascibus, 
prcecipue in crystallis, accompanies it with a letter in which, after 
claiming that he has extended some theorems of M. Kummer, he 
thus refers to Hamilton honourably but amusingly designated 
and to his prior results : ' But that my deduction for the special 
case (namely, the planes of vergency) may be correct, is clear 
thereout, that your great fellow, M. Hamilton, has already found, 
in a manner very different from mine, both principal theorems, 
that of M. Kummer and mine (Transactions, vol. xvii., p. 122, &c.).' 

At the annual meeting of the Eoyal Irish Academy in March, 
he seconded, with a gratifying tribute of appreciation, the pro- 
posal by Lord Talbot de Malahide that Professor Charles Graves 
should for the ensuing five years be the President of the Academy. 
On his nomination by the new President to be the Senior Vice- 
President he wrote to his friend the following not uninteresting 
letter. 

From SIR W. E. HAMILTON to the VERY EEV. CHARLES GRAVES, 

D.D., p. R.I. A. 

'March 18, 1861. 

* Although I shall, within an hour or two, have the satisfaction 
of seeing you in the Chair of the Council, yet I wish to state in 
writing that I accept with pleasure your nomination of me as the 
Senior Vice-President of the Academy, as I did the correspond- 
ing nomination of Lloyd on the occasion of his succeeding me. 
It was the first act of your prerogative and a graceful one at 
least I have no reason to suppose that the Academy will judge of 
it otherwise. At the same time you will perhaps pardon me for 
saying again that the acceptance of the office, although one of which 
I am well prepared to appreciate the importance, was designed to 
bo, on my part, a mark of the very great satisfaction, private and 
public, with which I saw you placed, on Saturday night last, in 
the Chair once occupied by myself, and both before and since by 
persons more deserving than 1 was. 






AI.TM. .14.] Honorary Degree at Cambridge. 131 

1 What I said in your presence to Dr. Reeves was, in essence 
though not in words, the admission of the natural feeling that 
having been for a while the first, it could be no object of amhition 
for me now to be the second in the Academy, yet I will say, on 
reflection, that it is a pleasure to me to be such for one year under 
you. For you will remember that when next year arrives, even 
if I should be then alive and well and on the Council, it will be in 
due course that my name should drop from your list of Vice- 
Presidents; and no member of the Academy, and least of all 
myself, could for a moment have imagined that any slight would 
have been intended, or conveyed, if you had lately used your pre- 
rogative otherwise than by appointing me. . . . ' 

An honour of a different kind summoned him in May to 
Cambridge. He received from the Vice- Chancellor, the Hon. and 
Rev. Latimer Neville, a notification that the Council of the Senate 
of the University desired to confer upon him, ' as a mark of their 
respect and esteem, ' the honorary degree of Doctor of Civil Law. 
Having been in 1845, after the meeting of the British Association, 
admitted by the same University to the ad eiuidem degree of Doctor 
of Laws, Hamilton at first felt uncertain whether he was free to 
accept the proffered honour. He thus states his difficulty in a note 
to his friend Dr. Anster : ' I remember kneeling as a Candidate 
Doctor of Laws before the Vice- Chancellor of the time : is it 
lawful that I should kneel again for a degree no higher ? Although 
perhaps in England it may be looked upon as such. Anne aliquis 
6/.s baptizari potest ? For it was a sort of baptism in the name of 
the Holy Trinity. For about sixteen years I have supposed my- 
self to be a Doctor of Cambridge, and certainly wore there the 
Doctor's gown, where it counts for something in visiting public 
places,' &c. 

Another consideration which weighed with him in his hesitation 
was his fear lest he might do anything to lessen the dignity of his 
Dublin degree. 

He laid these facts before the Vice- Chancellor, adding that as 
part of the heading of a communication (on Symbolical Geometry) 

K 2 



132 Life of Sir William Rowan Ham ilton . [ 1 86 1 . 

to the Cambridge and Dublin Mathematical Journal, published in 
1846, he had set himself forth as * LL.D., Dublin and Cambridge/ 
&c. The reply was : ' The ad ciindem Degree which you took is 
not the same t'n point of fact which we offer you, and therefore I 
hope you will be able to accept it.' The terms of his acceptance 
show how much he felt the honour. I extract them from his letter. 
* ... Under your sanction I now accept the offer of the additional 
degree of D. C. L., which I shall very highly prize. Indeed I can 
scarcely imagine or rather I am, at present, entirely unable to 
conceive any more gratifying compliment which could from any 
quarter be paid to me. . . .' His visit to Cambridge brought him 
much pleasure. The ceremonial in which so honourable a part 
was assigned to him took place on the 21st of May. The distin- 
guished men who shared with him the honour were Lord Stuart 
de Eedcliffe, the Earl of Elgin, Dr. (Ml (Bishop designate 
of Madras), Sir Eoderick Murchison, General Sabine, Mr. Grote 
the historian, and Dr. Eobinson of Armagh. He records that in 
the order of procession he came next after Lord Elgin, that he 
was in the speech of the public orator praised more than he chose 
to relate, and that before the banquet he was presented to the 
Prince of Wales, with whose manner giving the impression of 
receiving rather than conferring honour he was favourably 
struck. Murchison, Sabine, and Eobinson were old friends; 
and other old friends he found in Cambridge in Mrs. Eobinson, 
her daughter Mrs. Stokes, and Miss Mary Edgeworth, her niece, 
to whom were added Professors Challis and Adams. Dr. Whewell 
happened to be absent, and in a kind letter expressed to Hamilton 
his regret at missing him, and his pleasure that Lady Affleck had 
made his acquaintance. On leaving Cambridge, Hamilton took 
the opportunity of visiting what "Whewell, a connoisseur in Gothic 
ap'liit'-cture, called in his letter the * noble cathedral' of Peter- 
borough. 

He was again in his own Observatory on the 27th of \. 
and early in the following month had the pleasure of receiving 
there, for long converse during two days, his fellow-worker in i 



AETAT. 54.] Visit from Tail. 133 

Quaternions Professor Tait. The record in his journal of this 
congenial intercourse shows that it was much enjoyed by him. 
The books in preparation by both on their common subject Qua- 
ternions were well talked over. His journal of the first day 
records : < I walked with him nearly to the foot of the lane. At 
parting he wanted to be quite sure what my own wishes were on 
the subject of our respective publications. I said that they could 
litorally be expressed in two words " Sixty-one, Sixty-two : " 
meaning of course that / should have 1861 free to myself, and 
that he might have 1862 to do what he liked in. ' In a letter of 
1862 (August 29), Hamilton writes to Tait : ' You are perfectly at 
liberty to refer [in an intended contribution to a scientific periodi- 
cal] in any manner you choose to my forthcoming volume, and 
generally I have entire confidence in your discretion. The only 
thing I asked was that you would not publish a separate work 
before the appearance of the Elements. I shall be charmed, for 
both our sakes, to set you free as soon as possible.' 

This, and other similar statements, show how little able was 
Hamilton to forecast the time which his great work would occupy, 
or the extent to which it would proceed. I have already recorded 
that at first he imagined it would not exceed 400 pages, and now 
we see he felt sure it would be completed within the year 1861 
he had previously named a much earlier date whereas, in fact, 
the work extended to nearly 800 pages, and was left incomplete at 
his death in the autumn of 1865. His mistakes were attributable 
not to any want of industrious perseverance, which in him was 
habitual, but to the vastness of the scale upon which it was 
natural to him to work, and to his inability to calculate the 
extent of time which would be required for the mechanical trans- 
ference into manuscript and printer's type of the mathematical 
operations and conclusions which were already mentally conceived 
with perfected grasp and distinctness. 

I think it only just that I should here frankly state that a 
perusal of the correspondence between Hamilton and Tait has 
left on my mind, first, a conviction that, in tin's matter of their 



134 Life of Sir William Rowan Hamilton. [isoi. 

projected works on Quaternions, Professor Tait acted throughout 
A\ith strict regard to honourable obligation, and, secondly, a corre- 
sponding impression that Hamilton, in desiring his scholar to with- 
hold the publication of his Examples in Quaternions until after the 
appearance of the Elements, may have fallen into an error, though a 
most natural error, of judgment. Besides miscalculating the time 
which the completion of his own work would require, Hamilton 
naturally felt it to be scarcely fair to himself that one to whom he 
had freely communicated unpublished methods and results should 
precede him in making them known to the scientific world ; and, 
looking beyond himself, he felt desirous to secure for his own 
University the credit which would attach to the first publication 
of a methodical treatise on Quaternions. It may, however, I 
think, be believed that, had he not been too modest to estimate 
fully the magnitude of his own undertaking, he would have been 
aware of its superiority to all possible rivalry, and would have 
arrived at the conclusion that the earlier publication of Professor 
Tait's Examples would have tended, not to injure, but to promote 
the success of the Elements, by keeping alive an interest in the 
subject, and by preparing students for a proper comprehension of 
the magnum opus that was to follow. Professor Tait's work was 
jtublished, after that of his master, in 1867, and an enlarged edi- 
tion, in producing which he enjoyed the co-operation of Professor 
Kelland, in 1873. This edition bore the title of Introduction to 
Quaternions, with numerous Examples. 

The following month of July was rendered notable to Hamilton 
by the appearance of the great Comet of the year, first seen by his 
son Archibald, amongst its earliest observers, on Sunday, the 
'i"th of June, and then on succeeding nights watched for by 
himself. 

He notes, July 4, 1861 : ' It is certainly a very fine ono and 
probably much superior to Donati's [1858], although I have not 
yet seen it favourably. But I was greatly struck by the bright- 
ness of its nucleus, as seen through clouds, at 2 o'clock on $ 
morning. 



\i TAT. 54.] Social Science Congress. Ingleby. Kirk man. 135 

On the 10th of the month the Astronomer himself, in an 
environment unusual to him, appeared before the public as pre- 
sming over a lecture, given at the Rotunda in Dublin, upon 
.Taj an by Dr. Mac Gowan, who had been a missionary in Japan 
and China, when these countries were less known than they now 
are, and whom Hamilton had met in 1859 as one of the guests at 
Banchory House, near Aberdeen. Hamilton also, in the succeed- 
ing month, took some part in the public meetings, over which Lord 
Brougham presided, of the Social Science Congress. And he had 
the gratification of introducing to Lord Brougham his second son 

i student in physical science, who had manifested original 
thought in investigations connected with earth-currents of electri- 
-ity. But the greatest gain brought to him by these meetings was 
his friendship, then first entered upon, with Dr. Charles Mansfield 
Ingleby, who, taking advantage of his visit to the Congress, sought 
to be allowed to make the acquaintance of one whom he said that 
Englishmen regarded as the greatest of living Irishmen. Though 
accomplished in both branches of science, it was as a metaphysician 
rather than as a mathematician that Dr. Ingleby approached 
Hamilton. Their day of meeting was occupied with discussions 
on Kant and his philosophy discussions which led to a correspon- 
dence from which some extracts will be given. Shortly after- 
wards, contended for as a guest by friends new and old, and this 
time received by Mr. Oliver Heywood, Hamilton attended the 
Meeting of the British Association at Manchester, and on his 
return visited, at Croft Eectory, near Warrington, a brother 
mathematician, Thomas Pennington Kirkman, a graduate of 
Trinity College, Dublin, already favourably known to him by 
correspondence. That Mr. Kirkman was a labourer of thorough- 
ness and perseverance akin to Hamilton's own may be inferred 
from the following passage in a letter written by him to Hamilton 
in 1862. His field of investigation was 'polyhedra,' the same 
which had yielded to Hamilton his Icosian Calculus. Speaking 
of his results, Mr. Kirkman writes : 

' July 15, 18G2. The labour you may judge of when I say 



136 Life of Sir William Rowan Hamilton. [i> 

that the calculations cannot be written out in fewer than 800 close 
quarto pages. I have 500 pages of reticulations written, namelj, 
the preparatory list of groups of reticulations that can be made oat 
of nine particularized polygons charged with polyhedra of nine 
edges and under ; and I have not by a great deal completed eren 
these, in which I do not go beyond constructions of twenty-four 
edges.' 

The following extracts will prove to the reader what were the 
estimates formed by these men of each other. Kirkman writes : 



15, 1862. I wish I had the good fortune to be nearer 
to such a mathematician as you ; for it would be of immense ad- 
vantage to have the profit of conversation with such a man. It is 
a great loss to me to live cut off from all scientific intercourse/ 

Hamilton to Kirkman [from a draft], July 28, 1862. . . . 'It 
would be very difficult for me to express, without having the air 
of flattering, how much I admire your mathematical genius and 
discoveries ; but it is a real pleasure to me to be allowed any little 
opportunities, such as these, of testifying my respect and good- will. 
Some time or other I hope that you and Mrs. Kirkman may be 
guests of Lady Hamilton and myself here. But it is only fair to 
apprise you that my wife, like yours, uses an " Arabian " for her- 
M-lf, and that we have no carriage, &c. In short, I lead a very 
retired life ; which, as I grow older, I devote more and more to 
study. I might now find a difficulty in getting up a dinner-party 
for you, though I have given a good many in my time. Perhaps 
I may just be allowed to add that I am conscious of not having 
been in my normal state of health during the whole of my last 
visit to Manchester. . . .' 

"What Hamilton meant by devotion of life to study is indicate! 
by facts which I find incidentally recorded, that on one day in 
August, 1861, he spent * more than twelve hours at work,' and on 
the 29th of October following 'at least thirteen consecutive hour*.' 

At the end of the year Hamilton indulged himself with tlu 
purchase of 7."" ////' Greatest Enchantment, fyc., from the Spanish 
of Calderon, by D. F. MaoCarthy. In perusing this translation, 



. 54.] Calderon. 137 

specially commendable for its adherence to the metre of the 
original, Hamilton came upon a passage expressing the connexion 
between philosophy and mathematics, which as a philosophical 
mathematician he could not but challenge as a mistaken interpre- 
tation. He thus, with characteristic deference, submits the point 
to the translator, whose reply was a frank admission of the blot 
hit by his friendly critic. 

From SIR W. R. HAMILTON to DEMS FLORENCE MACCARTHY. 

* OBSERVATORY, December 13, 1861. 

* You have now a great opportunity of instructing me at once 
in Mathematics, Philosophy, and Spanish. 
' At p. 43 occurs the passage 

" No te digo, que estudie 
Con generoso motivo 
Matematicas, de quien 
La nlosofia principle 
Fue ; no te digo, que al cielo 
Los dos moviinientos mido, 
Natural y rapto, . . ." 

which of course increases my intellectual respect for the Enchant- 
ress of ancient fable. I suspect that Ulysses hardly caught her 
allusion to the distinction between natural and violent motion ; 
but, waiving Astronomy, and all old speculations on physical 
science, I want to ask you whether Circe may not (through her 
interpreter, Calderon) have spoken of Philosophy as the base or 
" principio " of Mathematics, rather than the latter of the 
former ? . . .' 



138 Life of Sir William Rowan Hamilton. [1862. 



CHAPTER XLVII. 

EXPENSE OF PRINTING. ROBER's ESSAY. AMOUNT OF SCRIBE-TOIL. 

(1862.) 

AT this time, when his Elements were approaching the five- 
hundreth page, Hamilton was brought into a state of serious 
anxiety by the exhaustion of the grant which the Board of Trinity 
College had made towards defraying the cost of printing. He saw 
that he was threatened with an expenditure which his private 
resources were insufficient to meet. In this crisis he imparted his 
difficulty to his friend Dr. Hart, from whom in return he received 
a suggestion that, following the successful example of Dr. Salmon, 
he should try whether an English publisher would not share with 
him the risk of publication. From letters of Hamilton written to 
Dr. Hart on this subject I make some extracts which will be 
found not only to unveil the extent of his anxieties and his labours, 
but incidentally to express in memorable terms what were his 
feelings towards Dr. Salmon, and what were his convictions in 
regard to the work he had in hand, and to the need that, in using 
and carrying forward his discovery, he should have the aid of 
other labourers in the field of the Higher Algebra. 

From SIR W. E. HAMILTON to A. S. HART, LL.D., S.F.T.C.D. 

'3fay 17, 1862. 
To you I shall then send a fair and final copy of 



half-sheet, but when I may have another half-sheet of the 
to send you is more than I can say. For the crisis is come at last, 
and the impression is suspended for want of funds. I have not 
yet made any formal application to the Board, much less to tho 



AETAT. ,~).~t.~\ Expense of Printing. 139 

Boyal Society, or any other quarter, but have forwarded all the 
pages to my old friend, Dr. Lloyd, who, as you told me, acts now 
as Kegistrar. Faithfully yours.' 

* P. S. I know that you are not the officer of the College to 
be troubled just now with any application ; but since we have been 
lately in free communication with each other, I shall just say that 
I think the Board would do themselves honour, and serve the 
cause of Science, by voting a new (and final) grant of 100, to 
make possible the publication of the Elements. It is all fair that 
I should run some risk say even 50 as probably I shall after 
that ; but has Ireland ever produced a new branch of Mathematics, 
or, say, only a new Calculus before? And can you or anyone else 
really suppose that I can enter the market on any terms of equality 
with our friend Salmon, whom I profoundly admire and love but 
who finds Ms public ready made, namely, all the mathematicians 
of the living world : whereas I have to create my public ? In 
short, can you seriously suppose that the publication of the Ele- 
ments is a commercial enterprise? ' 

' May 24, 1862. . . . The chief feature of the whole work- 
whenever it can be printed and published will probably be that 
Theory of Linear Functions of Vectors and Quaternions, to which 
I lately alluded. As you have been so good as to seem to take 
some interest in it, and in the work generally, I have thought that 
you might not dislike my making up for you what I consider as 
my final copy of the first six articles, with their sub-articles, of the 
sixth section of the last chapter. You need not be in any haste to 
return it, at least until it can be printed, and you can let any 
friend in College see it if you choose. It will show, I think, that 
great pains have been taken to produce clearness and compression. 
Yet I fear that the copy sent represents only about a third part of 
the section. After it, however, there will remain only applications, 
easier by far, and which it will be the distinctive mark of success 
to be able to treat briefly. Besides, Application enters only 
subordinately into the conception of my present work. I want to 
finish a Book of Reference a short one, unluckily for sale, it 
cannot now be but my intention for myself, and hope as regards 
other writers, is that the Elements may be cited, almost like the 
orotxaa of Euclid, in future treatises or memoirs on the Quater- 
nions. My own hope is to cite them in future essays in the Trans- 
actions of the Eoyal Irish Academy, &c. ' 



140 Life of Sir William Rowan Hamilton. [1862. 

* May 27, 1862. ... I have wished to submit to the Board, 
previously to applying for their aid, the manuscript of that sixth 
section of the final chapter on Linear Functions of Vectors, &c., 
which will be, as I expect, the principal feature of the volume, 
and of which the geometrical, and, above all, the physical applica- 
tions appear likely to be of vast importance in the hands of other 
persons. ... It would be rash to promise that the volume shall be 
concluded under 500 pages at least if some contents and some 
preface be included but I assure you that I am beginning to 
wish to have done with it. The labour has been very severe, 
including the mechanical part, which, however, appeared to be 
necessary. I estimate that the work will have cost me not less 
than ten thousand hours. Let us call it the advance (and not the 
c of the Ten Thousand.' 



On the 14th of June his old friend Dr. Lloyd had, as acting 
Registrar, the gratification of communicating to Hamilton the 
consent of the Board to advance a second 100 towards the print- 
ing of the Elements, accompanied, however, by the conditions that 
Hamilton was, as he himself had proposed, ' to pay the remainder 
of the cost, and that the foregoing sum was to be paid by the 
Bursar when the work was completed.' In returning his best 
thanks through Lloyd to the Board, Hamilton adds : ' It will make 
the publication of the Elements of Quaternions in a satisfac- 
tory form possible ; and it is my business to look to the rest. It 
will protect me from actual pecuniary loss, and that is about 
as much as I look for except in the way of fame for the College 
and myself.' 

It must here be noticed, what the reader may have anticipated, 
that the continually increasing size of his book inevitably renewed 
the anxieties which I have been recording, and which were not to 
cease before his death. 

The following extract from a letter to myself (May 20, 1862) 

proves that, notwithstanding these anxieties, he was at this time 

1-iiig with unremitting zeal, and places on record his judgment 

as to the comparative merits of his two great volumes on Quater- 

nions : ' . . . I am still intensely occupied with my new work, the 



AETAT. 55.] Roberts Essay. 141 

Elements of Quaternions, which I consider to be incomparably 
superior as a book to the Lecture*, although the earlier one may 
perhaps be considered to possess a greater interest in the History 
of Science.' 

In the autumn of 1862 Hamilton's work on the Elements was 
agreeably interrupted by a scientific investigation of a completely 
different character. The Archbishop of Dublin, Dr. "Whately, 
had encouraged his son-in-law, Mr. C. B. Wale, to act upon the 
wish of a friend by requesting Hamilton's opinion on two essays 
by Friedrich Grottlob Eober, the object of the first of which 
was to set forth a geometrical construction of the regular hep- 
tagon. 

The circumstances and the result rendered this a very interest- 
ing investigation, for the supposed geometrical discovery announced 
had been arrived at by Rober, who was Professor of Architecture 
at Dresden, from considering the measurements of an Egyptian 
Temple that of Edfu on the Nile. And so much of interest still 
attaches to the question of the extent and purpose of the use of 
geometrical knowledge in the structure of Egyptian Temples and 
Pyramids, that I think it right here to insert two of Hamilton's 
original records of his investigation.* 

It is true that his results were published in the Philosophical 
Magazine for February, 1864 ; but my reader will, I doubt not, 
enjoy the perusal of documents written when the impressions 
made upon his mind by the study of this interesting subject were 
fresh and deep. I reproduce, first, Hamilton's precis of the facts, 
as I find it in Manuscript Book B,, 1845, p. 121, and, next, his 
letter to Mr. Wale, which gives glowing expression to his feelings, 
as well as a pointed statement of his results. 



* In discussing the question to which I have ahove referred it has, I think, 
to be borne in mind that although these Egyptian edifices may have been, as to 
their main objects, religious or memorial, their architects undeniably exhibited 
in the construction of them a notable amount of geometrical knowledge, and may 
not unreasonably have been supposed to have adopted means of indicating and 
so conserving the truths upon which they worked. 



142 Life of Sir William Rowan Hamilton. [1862. 



* ON ROBER'S CONSTRUCTION OF THE HEPTAGON. 

1. On September 11, 1862, Mr. C. E. Wale, son-in-law of 
the Archbishop of Dublin, drove out to this Observatory, but did 
not come in : he left, however, a note to me, and also a couple of 
German Essays, by Friedrich Bober, which had been sent through 
Mr. "Wale, from Mr. Sharman, of Montreux, Canton de Vaux, 
Suisse, with a request that I should examine them, and give my 
opinion on their accuracy and value, or, at least generally, send 
some message by him to his friends in Switzerland, on his return. 

' 2. The earliest of the two Essays is entitled : Beit rage zur 
Erforschung tier Geometrischen Grundformen in den alien Temp I > ,i 
Aegyptens, und deren Beziehung zur alien Naturerkenntniss, con 
FRIEDRICH ROBER. Mit IV Lithographirten Tafeln. Dresden. 
Verlag von Voldemar Turk, 1854. 

'3. I understand the author to say, in his preface, that his 
deceased father, who seems to have been Professor of Architecture 
in the Academy of Dresden, and to have died at Paris in 1833, 
conceived himself to have explained the construction of the ancient 
Temple of Edfu, as being connected with the Inscription of a 
Regular Heptagon in a circle, on a plan which he had discovered. 

* 4. Accordingly, at page 15 of the Memoir or Essay, he gives 
but I take it as a process discovered by his father a geometrical 
construction for the following problem : " Einen gleichscheuk- 
ligen Triangel zu beschreiben in welchem jeder der Winkel an 
der Grundliuie das Dreifache des dritten Winkel ist." And the 
construction is illustrated by a Diagram, Tab. i. Fig. 1., which is 
not of excessive complexity. 

* 5. My first impression was that I should find either an error, 
or at least a rude approximation ; but to my great surprise, indeed 
astonishment, on repeating all the calculations, with Taylor 1 ** 
Logarithm^ I found myself unable to decide whether the result 

d in excess, or in defect: assuming that it must err, because 
only right lines and circles are employed, while the problem depends 
essentially on the solution of a cubic equation, and therefore cannot 
be resolved by extraction of square roots alone. 

* 6. And I wrote a note, dated September 12th, to Prof 

De Morgan to express this feeling of surprise : giving at the saim* 
time the formula by which I had expressed the construction. 



AKTAT. 55.] Roberts Essay. 143 

1 7. Finding that seven-figure logarithms left the question doubt- 
ful at least as I had used them I resolved to go through the 
entire calculation by arithmetic alone, and without any tables. 

*8. I have since performed the calculation with still greater 
accuracy than before : aiming indeed at having 15 decimal places 
correct, and using often 16 decimals in the calculations. Content- 
ing myself, however, with 13 in the final result, I find that ltdber's 
construction gives 

- 1 = cos y- = 0-62349 00759 241 ; 
while the cubic equation 2x* + x* - x = J gives 
x = cos y = 0-62348 98018 587. 

' 9. Eober's cosine is therefore greater than the true one, by 
about 247 in the 9th place of decimals, which answers to a defect 
of arc, amounting to about 35 in the 8th place, if the radius be 
taken for unity and consequently if a tenth of a second be repre- 
sented by about 5 in the 7th place of decimals. His error then, on 
a single seventh of the circumference, is only about - 0"-07 ; and the 
error of his sevenfold arc is a defect, but one which scarcely amounts to 
half a second : or that sevenfold arc is indeed shorter than the true 
circumference, but only by about the 800,000^ part of the diameter. 

' 10. On the equator of the earth the error of a single seventh 
would only be about seven feet. If the radius were fifty feet, or 
600 inches, the error of the single arc would be only three halves of 
the thousandth part of an inch, and therefore scarcely visible, if 
at all, to the naked eye. On the great circle of this Observatory, the 
sevenfold arc would err by only about the eight thousandth part of 
an inch.' 

From SIR W. E. HAMILTON to C. B. WALE, ESQ. 

1 OBSEBVATORY, September 15, 1862. 

* A wish to gratify the archbishop and yourself was the first 
motive for my attempting to examine to some small extent the 
Essays of Eober which you had the goodness to leave for me a 
few days ago, and to form some opinion on their value, unimpor- 
tant as that opinion might be. But the Memoir on the ancient 
temples of Egypt (Eober, Dresden, 1854) has interested me 



144 Life of Sir William Rowan Hamilton. [isr>2. 

profoundly. Indeed I have scarcely been able, since I opened it, 
to attend to anything else ; and it led me into some long calcula- 
tions which I have only just completed to my satisfaction. As I 
have paid no special attention to Egyptian Antiquities, nor medi- 
tated much on such mystical guesses as some have made at their 
inner meaning, the only point which I could hope to study usefully 
was the geometrical discovery announced in the first memoir, namely, 
" the construction of the regular heptagon," which the elder Eober 
Appears to have divined, from the study of the ancient Temple 
Architecture.* 

* I entered on the subject, perhaps with prejudice ; for like 
most (if not all) modern geometers, I have been accustomed to 
hold, and indeed still do hold, that it is impossible to construct such 
a heptagon with the "right line" and "circle" alone. Yet, to my 
great surprise, I found no error in Rober's numbers ; and on re- 
peating the calculations on another plan, with Taylor's seven- 
figure logarithms, I found myself quite unable to pronounce 
whether Eober's arc erred in excess or in defect from the exact 
seventh part of the circumference; for that it must err I felt 
assured. 

4 It seemed, therefore, worth while to go much more closely to 
work ; and laying tables entirely aside, to perform the whole of the 
work for myself, by arithmetic alone, and especially by extractions 
of square rooty. And to be quite sure of a high degree of accuracy 
in the final result, I made it a rule to work with not fewer than 
fifteen decimal places, besides employing all verifications that I 
could think of in the progress of calculation, which, thus labori- 
ously conducted, has covered many sheets of paper, and cost me 
many hours on two or three successive days. 

* At last, however, it is finished; and I should have no hesita- 
tion to commit myself publicly to the result, which is, technically 
expressed, that the natural cosines of the angle assigned by Hobci s 
construction is, to thirteen decimals, 0*62349 00759 241, whereas 
the true cosine of the seventh part of four right angles deduced to 
a corresponding accuracy from a known cubic equation I find to 
be a Ittt I.' /m, namely, 0-62348 98018 587. Admitting, though I 
do not believe it, that the two or three last of these decimals may 



'1 ho subject of the Second Essay was The Pyramids and Fartli. 






i. 57.] Rober"* s Construction of tJic Heptagon. 145 

be wrong after all the precautions taken, I am quite satisfied that 
the cosine of the Eyi/ptian Angle for really Kober seems to make 
it likely that the Egyptians did employ it is somewhat yrratcr, 
and iliat the cosine of the true or geometrical angle (for of course 
we can in geometrical conception- divide the circumference into any 
number of equal parts) is somewhat less than 0*62340, and con- 
sequently that the supposed Ey>/)>tian rule of the heptagon is no' 
mathematically perfect, though liober seems to suppose it to be so. 
And if you should ever think me worth citing on the subject, I 
request you to bear in mind that such is one of the results of my 
investigation. 

* But now let us turn the tables and inquire how near does the 
supposed ancient rule come to the truth ? How small, in practice, 
is the error which theory pronounces to exist ? And I answer- 
that in practice the error does not exist at all. I do not think 
that experiments of measurement, &c., could be so conducted 
by men, at least in the present age, as to prove to sight that there 
was any error. For practical purposes, then, the elder of the 
Robers, or the old Egyptian sage whose secrets he supposed him- 
self to have divined, has done the impossible. 

'We have in this (Dublin) Observatory a circle movable in 
azimuth, the largest of the kind in the world. If on this Great 
Circle (and of which the radius is four feet) seven successive 
Egyptian arcs were set off from an assumed zero or initial point, 
the last would end within about the eight-thousandth part of an inch 
of that point with which i\\Q first began ; and on the earth's equator 
this sevenfold error, which is one of defect, would only amount to 
about 50 feet. In short I compute that the sevenfold arc falls short 
of the true circumference, but only by about half a second. And 
although we aim at not -neglect in y half or even the tenth part of a 
second in astronomy, yet such an error of observation very often 
occurs, and nobody could be sure of detecting the eight-thousandth 
of an inch, so as to say that it had not arisen from some slight 
*///; of the compasses, &c., in the construction. But calculation 
carefully conducted is infallible, and I have no shadow of doubt 
upon the question. 

* Yet the practical success of the rule is to me absolutely wonder- 
ful : and it is long since any discovery in science produced in mo 
such a sensation of surprise. It enables me more than before 

VOL. in. L 



146 Life of Sir William Rowan Hamilton. [I8fljj 

to realize what we are told of the "wisdom of the Egyptians." 
Perhaps the construction was disclosed to Moses though one 
cannot see what use it can have been to him, except perhaps as an 
element of intellectual preparation. Some think that Euclid who 
certainly knew much more than what was in his ElrnicHf* de- 
rived instruction from the East or from Egypt. If so, and if tin- 
rule was at that time known there, according to the Theory of 
llober, I do not think it necessary to suppose that Euclid refrained, 
from any oath or obligation of secrecy, from publishing the con- 
struction of the Heptagon. He was not in the habit of Jn-Jii -////// 
///* eycx in Geometry. It was nothing to him that the construction 
was reasonably simple, and that he could sec no want of " closing 
up " when the sevenfold arc had been completed. He doubtless 
asked for & proof : and the Egyptians had none to give.* Such 
then are my two maiu conclusions, which I may thus recapitulate : 

* (1) The (alleged) Egyptian Rule for the construction of the 
Regular Heptagon is, in rigour of theory, erroneous. 

* (2) The same rule of construction of the Heptagon is, however, 
for all practical purposes, perfect. 

'No artist of the present day, I feel sure, would undertake f> 
divide a circle into 7 (seven) equal parts, with a superior, or even 
an cytuit accuracy, to that which the construction, if fully carried 
out, would give. 

* I beg that, when you have the opportunity, you will convey 
my best thanks to Mr. Sharman of Montreux for his very welcome 
present of those two Memoirs of F. Itober. And I hope that by i 
the labour which I have recently gone through, on the occasion of 
reading one of them, and by the length to which this letter has 
extended, I have sufficiently marked my thanks to you, and my 

ionate respect for the Archbishop. . . .' 

' 11 million here gives to Euclid a character which belonged also to himself. 
I n member a comment of his upon a fact mentioned in a letter from Dr. 
HaujJ.ton, S.F.T.('.J)., now President of Hie Koyal Irish Academy a man in 

id to whom 1 have met with a strong expression of Hamilton's ' respect and 
admiration.' That litter contained the following sentences: '1 know of no 
11 who lias not seen Conical K. t'i.icli<.n that really believed in it. 1 have 
jn\ > li' conv r i a score of mathematicians, by showing them the Cone 

<>i J.n/lit.' Hamilton's laughing remark on the persons referred to ly Dr. 
Haughton was tin-: ' II. .w dili.n-nt from me! If I had mm it t.nl\. I 
diould not have heli.\,d it. My eyes have too often deceived me, 1 l ! 
because I 1 i it.' 






. 57.] Homers Method. 147 

In carrying through this investigation lie found it convenient 
mploy what he calls* 'that powerful method of llorner, for 
(lie arithmetical calculation of numerical equations/ a method In- 
had long known theoretically, but had not practised.! Of this 
in. '(hod Professor De Morgan was an intense admirer, considering 
it the greatest mv>,vA,/ discovery in arithmetic, lie furnished 
II imilton with some improvements in the manner of working 
with it, and in a sort of rivalry they both proceeded to calculate 

22 places of decimals the cos. of --, which, working indepen- 
dently, they found to be 

= 0-62348 98018 58733 53052 50. 

Hamilton's manuscript books afford striking evidence of the 

nut of numerical computation requisite to arrive at such a 

lit : they show also that he was not content with working upon 

the particular problem before him, but that other cubic equations 

furnish i i d him with material to which he vigorously applied the 

no method : in one case carrying out his calculation to 28 places 

I'-cimals. Yet this laborious work was in a sense play to him. 

< )n the 17th of September he writes to De Morgan : ' All this has 

.to distracted me from the Elements' On the 26th he writes : 

I must now drop arithmetic for awhile ; though the work gone 

through has rather refreshed than fatigued me.' In the autumn 

of tin' following year he again corresponded with De Morgan on 

is subject, and prepared the memoir respecting it, which was 

printed in the Philosophical Jftu/dztiH' for February, 1864 his last 

contribution to that periodical. 

It may here be mentioned that at this time Hamilton was a 
very solitary labourer. Except what the Observatory assistant, 
Mr. Thompson, could but rarely render him, he had no help in 



' New Method of solving Numerical Equation** l>y \V. G. Homer: I.on- 
'1-u, 1819. 

f See Munusciipt Book, ' ISGo, Number <V j>. '21, draft letter to ]\i* >..ii 
Archibald. 

I- -2 



148 Life of Sir William Rowan Hamilton. [1862. 

keeping copies of mathematical work or correspondence. His 
manuscript books had some years previously borne witness to the 
assistance dutifully rendered to him in this way by his two sons ; 
but now both of them had left their home. His eldest son had, in 
1862, accompanied his Aunt Sydney, his father's only surviving 
sister, to Nicaragua, where for some years she devoted herself tr> 
keeping a school for girls at Grey town,* and where he sought for 
employment ; and the younger son, who, for some time after his 
ordination in 1860, had served as curate of his native pari>lu 
Castleknock, and while so acting resided at home, had now 
entered into clerical engagements at a distance. Occasionally, 
indeed, the feminine hand of his daughter appears in those 
manuscript books, requisitioned on an emergency to copy a letter 
or even a scientific paper, but usually the labour is all his own, 
and it must have been enormous. Few persons, it is true, would ' 
have kept sucli full record of correspondence, and all particulars ! 
connected with it, as Hamilton was in the habit of doing, but still 
the great mass of this voluminous material was genuine and im- 
portant work. And it causes to the reader who examines it ; 
perpetually renewed feeling of astonishment at the amount o 
industry it represents. 



* It appears, from an interesting and able letter from her to her brother 
that Sydm \ Hamilton at this time desired much the rehabilitation of the 0! 
vatory at Bogota in Columbia (of which the site was the nearest to tli 
and at the highrst altitude of any Observatory in the world), and even ei 
tained tin- idi-a of ottering her own services towards the undertaking, as accus- 
tomed to oliMTve, and to make reductions of Astronomical observations. 



MM-AT. .>?.] Quaternions and Physic*. 149 



CHAPTER XLVIII. 

SCIENTIFIC ITEMS. CONCLUSION OF DE VERB CORRESPONDENCE. 
THE ROYAL IRISH ACADEMY AND THE ROYAL DUBLIN SOCIETY. 

(1863.) 

THE year of Hamilton's life at which we have arrived was marked 
by no outward events in his history, yet the reader will, I think, 
lind interest in some extracts which I proceed to give from letters 
which passed about this time between Hamilton and various scien- 
tific friends. They will be followed by the concluding portion of 
the correspondence with Aubrey De Vere. At the end of the 
chapter will be found Hamilton's * Memoranda ' of the part taken 
by him in an important discussion at the Royal Irish Academy on 
reposed subordination of the Academy to the Royal Dublin 
Society. 

From PUOFKSSOK P. G. TAIT to SIR W. It. HAMILTON. 



, 1862. 

'I showed Conical Eefraction to twenty of my class this 
morning (at once), having splendid sunshine.' 

* Aiujmt 26, 1862. Professor Bolzani is here and we have had 
long discussions on the subject [Quaternions]. He is immensely 
interested in it, and lias given a course of lectures on it in Russia, 

1 .. li k'h he has promised to send copies when printed. He intends 
to visit you in Dublin before returning to Kasan.' 

After urging Hamilton further to develop particular results, 
which he indicates, Tait continues : 

4 Such things as these would show the enormous value of qu.i- 
; lions to Phyxicixts, and that is in my eyes worth any amount of 
pun- analysis or geometrical theorems.' 



150 Life of Sir IVi Hi am Rowan Hamilton. [1863. 

Hamilton to Professor Tait, acknowledging the receipt of a ' 
proof of Chap. II. of Thomson and Tait's Dynamical La-* 
Principles, writes: 

*I am much ohliged, and expect to learn much from a 
careful perusal of the Treatise. And indeed I have much to i 
Leant. 

* The world of Science seems to admit that I had not only read 
but written to some purpose on Dynamics about thirty years ayn ; 
but a new generation has arisen. Energy and Work, in the old \ 
English meaning, are things not unfamiliar to me. But I li 
only the dimmest views of the modern meanings attached to those 
trims. From "Thomson and Tait" I hope at last to be instructed 
on the point: not, of course, without patient study on my part. 
A certain humility and teachableness of the ethical kind I can 
promise. 

* I have long got so far as to understand the foot-puinnl us 
applied to the raising of weights. But if 10,000 foot-pounds of 
work have been expended in raising 100 pounds through 100 feet t j 
and if the load be then let fall, and descend to its first level, I wish 
to be instructed whether and why and in what sense you Eneru 
deny that a negative myriad of foot-pounds of work has been clone 
by (j rarity. When poor Sisyphus had patiently toiled in rolling 
his great stone up the hill, and back again 

KUAIVOCro Aaae,' ai>ci(?}, 

(I quote from the Odyssey m< -morilcr, and probably inaccurately) 
might he not have consoled himself by the thought that the aven- 
jing Fates, or Furies, had been compelled to take some counter- 
acting trouble? Your book when complete, may enable even me to 
understand all this, slight as my instinct is for physics.' 



The same disposition to appreciate fully the work of others! 
and t<> Hibinit himself as a learner, while at the same time claim- 
ing the share of credit due to himself, is manifested in the follow- 
ing acknowledgment to its publishers of a presentation copy of a 
scientific treatise. 



T. ~>i.~\ Hamilton's Originality. 151 



To MKSSHS. MACMILLAN & Co. 



April W, 1863. 



4 1 am much obliged by the attention on the part of the authors, 
junl of yourselves, in your having lately sent me a complete copy 
of the obviously very valuable Treatise on Solid Geinm-try by Messrs. 
Frost & Wolstenholme. ... I have already treated, by my own 
calculus, a good many rather difficult questions of solid geometry ; 
for example, one which is analogous to, but greatly more general 
than, that of the Edye of a Tubular Surface, which I see treated at 
page 37 3 of the Treatise recently sent me. In doing so I inte- 
grate and interpret, by quaternions, a differential equation of the 
second order and second degree, between four variables, which was 
assigned by Monge in his Application de V Analyse a la Geometric 
(page 372 of Liouville's Edition, or page 325 of my old friend the 
1'Wrth Edition, Paris, 1809, with which I was familiar when a 
boy), but which was left reintegrated and ^interpreted, by the 
illustrious author of that great work. There will therefore be a 
vast deal to intercut me in the new publication which you have 
forwarded, and if it have arrived too late for me to learn much 
from it Just now, I hope at least to make an honourable mention of 
it, in any preface, however short, which I may prefix to my own 
work, for the publishing of which no arrangements have yet been 
imido. I am, dear sirs, requesting you to present my regards to 
the authors, your obedient servant/ 



Throughout the year 1863 Hamilton was engaged in the 
investigation by his own methods of osculating twisted cubics and 
in the discussion of a general centre of applied forces (subjects on 
which he made communications to the Royal Irish Academy, 
which were printed in its Proceedings, April 27, 1863, and June 22, 
1863), and he made frequent inquiries by letters to Dr. Salmon 
and Dr Hart as to whether his results had been anticipated. A 
postoript to one of L)r Salmon's replies was calculated to give 
him encouragement. 



152 Life of Sir William Rowan Hamilton. [!*>:;. 

1 October 1, 1863. You so seldom fall 011 other peoples' tnu-k, 
that when I met Hart a couple of days ago, and he told me you 
had been asking him whether certain theorems you had arrived at 
were new, I said : " Oh, tell him they are new. You will be quite 
eafe." ' 

In a letter of October 3, 1863, to his son Archibald (now, it 
will be remembered, a clergyman), who had been making some 
algebraical calculations, and sought for information respecting 
them, he supplies him with the computation of i [' e = 2-71828 . . . 
= the Napierian, or natural, base of logarithms '] by the exponen- 
tial series, and with its verification by squaring, in which he used 
a kind of contracted multiplication, which, he says, he picked up 
from his assistant, Mr. Thompson. The following passage has a 
value as evidencing the pains he habitually took to be accurate, 
and as conveying his advice on the point to his son : 

'I always verify the multiplications by divisions mentally, 
and add both upwards and downwards, often making a note of 
the figure carried to the next left-hand column. In general, I 
take all precautions for accuracy, and advise you to do the same. 
Whatever you do calculate and you know you need not do so 
more than you choose let it be carefully done.' 

Then after speaking of Horner's method, in terms which 1 
have already quoted,* he adds : 

' I shall gladly get you somewhat larger tables if you have any 
real occasion for them, only don't spend too much time on such 
things. How much can be spared you can judge for yourself. 1 
know that a complete change of mental occupation is occasionally 
useful and almost necessary. My book will get on none the worse 
for my having been lately thus working at arithmetic* 

His occupation with arithmetic was now unexpectedly pro- 
longed by his neighbour, Mr. J. G. Rathborne, submitting i> 
him a very meritorious system of contracted multiplication which 

* Supra, p. 147. 



\ in AT. 57.] Conclusion of Correspondence with A.De Vere. 1 53 



hud been devised by a self-taught man in his employment, named 
Boyers. Hamilton considered it to be ' new and ingenious.' * It 
had/ in comparison with a method ordinarily used by himself, 
'the advantage (he wrote, October 19, 1863) of giving a suc- 
cession of approximations, each better than the preceding, but 
required rather more of mental calculation in passing from step 
<o sl-p, yet not more than would be found easy by a practised 
calculator/ 

I may conclude these extracts with one from a letter written 
early in the year by the eldest son of Sir John Herschel. lie 
had paid a visit at the Observatory, and on his return home, in 
thanking for a letter of his father forwarded to him, quotes from 
it the following message relative to Hamilton's book : 

* When it is complete, as I hope it soon will be, I shall take it 
up ab initio, though without great hope of mastering it in the few 
years that remain to me after 72 ! But I anticipate for him (Sir 
William) a real triumph in its publication.' 

I now present to the reader the concluding portion of Hamil- 
ton's correspondence witli Aubrey De Yere, a correspondence 
which, in my estimation, constitutes an integral and most valu- 
able element in his life. I consider it a privilege to have been 
allowed to give it, even imperfectly, to the world, and am confi- 
dent that no reader of this biography would willingly lose the 
share contributed to it by Hamilton's friend. This friend paid 
him a visit at the Observatory in the early part of 1863. In this 
winter Hamilton was troubled by that inherited malady of gout, 
by which, as we have seen, he was visited at Cheltenham in 1856, 
which had subsequently recurred at intervals, and which was ere 
long to prove fatal to him. The first of the following letters 
refers to this disturbance of his health, and leads the writer to 
make interesting record of his experience in relation to the effect 
of bodily pain or illness upon the action of his intellect : 



154 Life of Sir Willia m Rent V7 ;/ Ha in ilton. [ 1 SG:J . 

From SIR "W. 11. HAMILTON to AUBREY DB VEUK. 

4 OBSERVATORY, February 9, 1863. 

' I enjoyed very much your visit, and my walk with you, and 
you may perhaps be glad to know that though (from want of 
recent practice) I felt since some stiffness in my legs, the feet have 
been quite uninjured, or, more plainly, no return of gout has been 
provoked. 

* I wish you would try a fit. For one's philosophy and temper 
it is an admirable exercise ; and to some constitutions it seems to 
lie tixrful. I had been wearying for a slight touch again, when 1 
was taken at my word last Christmas. 

* My intellect, such as it is, appears to me to be made evon 
clearer by sickness or bodily weakness, when such comes. The 
physical effect, however, is, that although I can scarcely spare 
more than a moment to attend to a twinge of the gout painful as 
that certainly is I cannot work, when an invalid, for so many 
hours consecutively, as when in full and normal health. A feeling 
of fatigue comes on. 

' But I call myself quite well now, thanks partly to your visit, 
and remain, my dear Aubrey, your old and affectionate friend.' 

The note to which the following is a reply has not come into 
my possession : 

From AUBREY DE VERE to SIR "W. E. HAMILTON. 

* 12, LEINSTKU-STKEKT, March 6, lsi;{. 

*. . . I have also a note to thank you for, parts of which relat ing 
to our conversation when we had our walk touched me VTV 
much. I can well understand how deeply you must have felt, 
being thus visited once more as by a hand stretched out from your 
boyish days. The heart never grows old if it ever was young ; 
and sometimes clings the more to the feelings that belong t<> its 
youth, when ynilh as counted by years is leaving us. I hav- 
Driven a reason for this in one of my poems called "Psyche," i 
. Affectionately yours.' 

The last letter written by Hamilton to Aubrey De Verc waj 
worthy of him as a friend and as a Christian; as a friend, who, 



\I:T.\T. ST.] Last Letter to A. De Vere. 155 



notwithstanding that serious differences of opinion had arisen 
between them, could still continue to love and trust his friend : 
;t Christian who, raised above minor details, could rejoice to hold 
last, in sympathy with his friend, the cardinal truths of his reli- 
gion. The last letter of Aubrey De Yere to Hamilton is, I think, 
one of singular beauty, both in the vision it calls up of the day.s 
that had gone by, and in the perfection with which thoughts of 
lender sadness and affectionate reminiscence are expressed. It 
binds together the early meeting of the two friends at Curragh 
( 'hase where Lord Adare and the parents and sister of Mr. De 
Vcro completed the party, and where social intercourse was en- 
joyed which was never forgotten by either with the days, now 
more than forty years afterwards, when their last words, words of 
faithful affection and Christian harmony, had come to be ex- 
changed. 

Front SIR W. II. HAMILTON to AUBREY DE VERE. 

' OBSERVATORY, April 3, 1863. 
< Good Friday. 

' MY DEAR AUBREY You may not think this day a good one 
for writing a letter even to an old friend; but I have just been 
reading again your Hymn for Good Friday, both in the edition of 
1 842, and in the poems published in 1855. Allow me to say, in 
passing, that if there be a word or two, in the latter edition, which 
you could not expect me to adopt, I yet admit that on the whole 
\ou have made it & finer poem. 

' It may be more important to remark that I humbly conceive 
myself to be as much a Catholic as you tccre, when you wrote the 
hymn in Us first form ; and it is a comfort, in these Colenso days, 
to have an opportunity of refreshing, by a reperusal of it, a sym- 
pathy so sincere in the most vital doctrines of that Christianity, 
which we both profess to believe : 

" Lamb of God ! on whom alone 

Earth's penal weight of sin was thrown, 

Have mercy, Saviour, on Thine own ! 

For thou art Man. The Virgin gave 
To Thee her breast, the earth a grave." 



156 Life of Sir William Rowan Hamilton. [1863. 

" Lamb of God, on whom was laid 
The debt all worlds had never paid, 
Have mercy, Saviour ; hear and aid. 
For thou art God. ..." 

" Thus, Christ, we turn from all to Thee ; 
Miserere, Doinine." 

'The " For thou art Man," and the "For thou art God," and 
the closing " Miserere, Domine," appear to be the moat practical 
teachings of the Christian Religion : or let me say, with greater 
reverence, among the most, if it be presumptuous and hazardous to 
distinguish. ... I remain, my dear Aubrey, your old and affec- 
tionate friend.' 

From AUBREY DE VERB to SIR W. E. HAMILTON. 



CHASE, ADAUE, May 14, 1863. 

DEAR HAMILTON I have not been as ungrateful as I 
may perhaps have seemed, in not having before now answered 
your letter written on Good Friday. . . . 

'Here I am, once more, after an absence of about two years, in 
my old home, and alone ; for my eldest brother and his wife arc 
in England. It is to me haunted ground. After a time, of course, 
this effect wears off ; but at first after coming here, it really seems 
to me a sort of enchantment. The present becomes almost nothing 
a mere vapour and the past becomes so distinct that I recognise 
the steps of the departed as well as their voices. The most trivial 
incidents rise up before me wherever I go; and in every room of 
the house, and every walk of the garden or woods, I see again the 
old gestures, expressions of face, even accidents of dress, which no 
one could fancy could have lived in memory. I allude of course 
to my 1'atln-r and mother principally, but not to them only. Very 
old friends, most of them long since dead, walk with them; and 
tin* eld jots an- repeated, but with a strange mixture of pathos 
and mirth ; and my brothers and sisters (there wore two younger 
sisters who di*<l each at the age of fourteen) seem to me once more 
oM davs of childhood or opening youth. 

'I suppose that many people must have had this experience. 
To me it seems nearly the most solemn and pathetic one that lit.' 
brings. With all its melancholy the preponderating feeling it in- 



AKTAT. .07.] Last Letter from A. De Vere. 157 

eludes is one of sweetness. As life goes on, and goes by, a certain 
tenderness, almost remorseful tenderness, seems to attach to all 
human relations and even to inanimate objects. We seem to be 
such mere shadows such a helplettneM seems to belong to us, and 
our parents, and all the fleeting generations that glided on and off 
flic stage of life, in days earlier than those we remember, that it 
appears an inconceivable cruelty that anyone should "agitate the 
light flame of our hours " by unkindness to a neighbour, or even 
l>y want of sympathy. It seems as if everyone must have been 
intended to be in some sense a "helpmate" to everyone else, and 
that to hurt instead of helping any of those who belong to so feeble 
ji race a race whose joys are so fleeting, and whose trials are so- 
many, irrespective of those which come from a fellow-being, im- 
plies that a long madness is preying upon human society. 

I am writing in the old library, and the writing-desk on which 
my father wrote all his poems and letters is the one my paper 
rests on. Among the Phantoms which have been visiting me is a 
Phantom visit from Adare. One of the party is a young Philoso- 
pher of whom we had already heard as a Scientific Wonder : and 
now from his discourse we find that he is as much devoted to Poetry 
and Metaphysics as to Mathematics. Then there comes the 
ramble in the woods, the departure, and my father's enthusiastic 
comments on the young Philosopher, and especially the praise 
bestowed on the " transparent candour " which made all his 
thoughts seem visible things. 

* Even those whom I never saw, but who belong to this magical 
Past, share in the strange spectral Resurrection. There is poor 
Francis Edgeworth. We used to talk of him and his poetry ; and 
as I walked about our lawn yesterday there came before me as 
freshly as ever his poem with the lines 

" It cannot soothe me in my loneness ; 
It cannot mitigate my oneness." 

* Such a strange beauty belongs to the Past that it seems to me- 
as it' half the value of the Present is derived from the knowledge 
that it will one day be the Past. It often happens that the 
moment a friend has left the room (especially at night when they 
separate till morning) he becomes invested with a new character ; 
and a softness falls on the image (that lately wore all the rough- 
ness of a jarring Present) from the distant Shadowland. Of all 



158 Life of Sir William Rowan Hamilton. [1863. 

the feelings of this sort, the strangest is that which comes to us 
from the recollection of our parents at an age when they were, as 
we then thought, getting oldish, but which, looking back on it, we 
see to have been Youth the Youth we have ourselves lost in turn. 
To look on them thus with filial reverence, and yet with thatjwp- 
tictive sadness with which we regard the young and inexperienced 
who are unconsciously losing what they must so often lament, is a 
mixture of feelings which admits of no name. It makes one 
realize the facts that even the Parents of our Parents, who seemed 
to us like Patriarchs, had their brief day too, and were but poor 
grentle wrecks stranded in the sunshine, when we knew them. 
The feeling thus inspired seems to me to help us to conceive that 
relenting tenderness with which the great Father must look on all 
His creatures, remembering that " they are but clay, " and that 
with all their high hopes they have been so mysteriously " made 
subject to vanity." However, the next World is the world of 
realities ; and here the shadows rehearse at least the great Future 
drama. For almost all of us there must surely be some interme- 
diate place where we are made capable of looking unblinded on 
the wonders of the Future. To me there always seemed some- 
thing so unphilosophical as to be almost incredible, in the assump- 
tion that death (which is not even a sacrament) should so act as a 
diarm or talisman, as at once to render the mass of even the good, 
who have hitherto been occupied mainly with the Shadowland, 
capable of sustaining the uncreated glories of the Beatific Vision. 
It is not a question of Guilt or of Merit, but of the time necessary 
for the development of moral habits and spiritual capacities, a- 
opposed to the notion of magical transformations. 

* I quite forgot that I had altered the hymn on Good Friday, 
till your letter made me refer to it. It is, I think, on the whole 
improved/ 

In December of the year 1880, Aubrey De Vere published in 
the Ii-i.-h Mont hi i/ MtHjdzine two sonnets * In memory of Sir William 
Rowan Hamilton,' which, as he says in a letter to me, 'show at 
least that Hamilton was one of those, one's friendship for whom 
will stand the wear and tear of some fifty years/ 

Of these sonnets I reserve one for a later page; the subject of 
that which follows 8ii^<-.st.s its insertion liore. 



AKTVT. .37.] A\ /. ~-'li'<tdc/iiv and R. I ). .W/f'/v. 

'IN MK.MOUV OF SIR WILLIAM ROWAN HAMILTON. 

'.\rn:K i;i;.\Di.\(i ACAIN ins LLTTKKS. 

' At times I see that jinij)le forehead lit, 
Bright as the day-spring round the mounted lark ; 
At times I sir thcc stand in musing fit; 
At limes iu woodlands oi' that twilight park, 
Deciphering well-loved names on beechen hark :* 
Where Kotha's moonlight ripples past thee tlit, 
I see thee kiss a gravi then by it sit f 
Hi-r grave that left the laud's chief Poet dark. 
This day I read thy letters. Word and scene 
Recur with strangely mingled joy and ruth ; 
Thy soul translucent, yet thine insight keen, 
Thy heart's deep yearnings and perpetual youth 
Thy courtesy, thy reverence, and thy truth 
All that thou wert, and all thou might'st have been ! 

l\1,ni,iry 20, 1880. 'A. I)E V.' 



Memoranda respecting the Meeting of $ night, July 6, 1863, 
of the Royal Irish Academy. [Manuscript Book L. 1863, p. 13.J 

'On D the 6th I went in the evening to attend the Special 
General Meeting of the Royal Irish Academy. That meeting 
had been convened by the President and Council, to take into 
consideration some proposed Resolutions on the subject of a 
Report recently published, of certain Royal Commissioners, who 
had, as the Council conceived, and as the Academy voted, sug- 
gested our being placed to some extent under the control of the 
Council of the lloyal Dublin Society. 

'The whole affair had been managed witli a most suspicious 
secrecy on the part of the Commissioners ; and an impression exists 
that there has been a job at bottom. A good deal of indignation 
praa expressed by several speakers, of whom I was one, coming 
next after the Treasurer, Dr. Carson, who followed the President, 
Deuu Graves. 

' The President stated, that as soon as he received (about ten 



Vol. i., p. 515. f Supra, p. 



160 Life of Sir William Rowan Hamilton. [1863. 



Hays or a fortnight ago) information of what had been recommend* <] 
by the Commissioners, he did not lose an hour in writing to tin 
Lord Lieutenant, to protest on the part of the Royal Irish 
Academy, of which he was the head, against any such recommenda- 
tion being adopted. And in my speech I expressed a hope that 
the Academy would vote their unanimous thanks, before the 
Meeting separated, to Dean Graves for his prompt, spirited, and 
useful action in the matter. Accordingly, when the Resolutions 
sent down by the Council had all been voted, with only sonu 
verbal changes in the two last of them, it was proposed an<l 
carried by acclamation, that the special thanks of the Academy 
should be given to the President ; further action being remitted to 
the Council. 

* I was led to state that I entertained only respectful and 
affectionate feelings towards the Royal Dublin Society ; and that 
I was not insensible to the honour reflected on Dublin, and even 
on Ireland, by a Society so ancient and so useful. 

* I valued much the compliment which had been paid me, 
many years ago, by my election as honorary member of that 
body; and had at least on one important occasion availed myself 
of my privilege to speak, although I could not vote, in the Socu-t \ . 

' And to show more completely how far I was from feeling any 
disrespect or unkindness in that quarter, I ventured to relate a 
circumstance which had occurred many years ago, and which I 
not in the habit of talking of, although at liberty to do so, becan-- 
I had taken no oath of secrecy on behalf of myself and fellow 
the time. 

* It had happened, then, about twenty-two years ago, that I 
was a member of a Commission which met at Mac Cullagh's rooms 
in Colle j ge, and to which the consideration of the affairs and const i- 
tution of the Royal Dublin Society was submitted. And it then 
appeared to me that I was at first the only member of that Com- 
mission who did not aim at the abolition of the Society, or at ! 

at a 1'u-nnxh'm-tiun so complete as practically to amount to abolition. 

' But I had been firm in my opposition to any such mca.- 
or recommendation, and had given it to bo distinctly undn>i 
that 1 would withdraw i'mm tin- Commission if such were t> 
made the basis of its Report. I prevailed ; and it had ever since 
been my feeling, although seldom if ever expressed, that / 



AETAT. 57.] R. I. Academy and R. D. Society. 161 

1 That Society was older than our own, which had not existed 
quite eighty years ; it was also richer ; and on each of these two 
grounds, but especially on the first of them, a certain pm-cr/curp of 
ceremony might cheerfully be conceded to it by the lloyal Irish 
Academy. It was not merely that the President of the Eoyal 
Dublin Society happened also to be the representative of the 
Sovereign : were he only a private member, elected to that Chair, 
" You, sir," I said addressing myself to Dean Graves " would 
not be humiliated by allowing him precedence on any occasion of 
state or (rrcwonifil, on the ground of that relative antiquity of 
the Institution which he was thus supposed to represent. We 
want no Ulster King, no Garter King-of-Arms, to settle that point 
of precedence. But if it be proposed to concede any other sort of 
precedence, or to allow any kind or degree of control to be exercised 
by the Council of the Eoyal Dublin Society over the Eoyal Irish 
Academy or its Council, then, sir, I must reject such a claim with 
an Indignation which I am sure that I do but share with every 
member present." 

' I do not now (6 July 7, '63) assert that those were my exact 
words last night, but I remember that I spoke in that tone, and 
made remarks to that effect. My speech was not a very short one ; 
it was animated, and (I think) well received ; but it was entirely 
extemporaneous. At one stage, having previously mentioned 
MacCullagh iu another connexion, I again introduced his name, 
and said that I could conceive how indignant he would have been, 
if he had lived to see a proposal thus made to subject to the 
control of any other Society the Academy to which he had de- 
voted so large a share of his gigantic intellect in science, and 
on which he had concentred so much of his deep love of country. 

' It was, I think, then that I made a transition to myself; to my 
own history and feelings. I said that I could remember the time 
when, being but a boy, and not yet aspiring to the honour of being 
a Member of the Eoyal Irish Academy, I at least looked forward 
to that of being permitted to write in its pages ; and felt, what has 
been my feeling ever since, that any contribution so received 
would be an offering laid upon the altar of my country. I could 
remember the pride with which I found myself elected a member 
of the Academy in 1827, and of the Council in 1828 ; to which was 
added the still higher honour, in 1837, of being called upon to preside 

VOL. III. M 



1 62 Life of Sir William Rowan Hamilton. [1863. 

over the body for a considerable course of years, with the unusual 
compliment at the close of such Presidentship of receiving a vote 
of thanks for consenting to remain, for some years longer, a mem- 
ber of the Council. And if I had not been included in the last 
list of auch members, I felt that no slight was designed thereby, 
but that it was known that I was engaged upon a scientific -. 
work, which would prevent my frequent attendance, and of the 
subject of which, although I might occasionally give some account 
to the Royal Irish Academy, I should certainly never think <>t' 
laying any such account before the Eoyal Dublin Society ; but 
if I could suppose that the Academy would submit to any such 
control as was proposed, I must ask to be forgiven when I said that 
I should then scarcely feel membership to be an honour. 

' I observed that we had no reason to apprehend on the part of 
the Government any attempt at coercion. The Academy and the 
Government for the time being had always been in friendly rela- 
tions. I had myself the honour, on four different occasions, to 
receive, as President, the Viceroy of the time ; and " you, sir," I 
tsaid to Graves, "have had some similar opportunities. We have 
no reason to apprehend hostility from any such quarter. Jlnf, 
if the worst comes to the worst, ice must not sell our Charter." 

* I then sat down, but remember that I had begun by stating 
that I had inquired whether there were any programme, or plan 
previously arranged, according to which the order of the speet 
was to be regulated ; for that, if so, I would have contented my- 
self with a few words, before the close of the meeting. Considering 
the whole occasion as an exciting one, I had been anxious to say as 
//'/ words as possible, and had in fact kept myself in a cohl Inf 
)niithrinat'u-K all the morning, that I might come to the subject as 
calmly as I could. It was only a very few days ago that I had 
heard anything at all respecting it ; and I might mention that last 
Saturday, when I happened to have some conversation respecting 
the proposed resolutions with a distinguished intellectual man [it 
was L)r. Lloyd, after the Visitation], who happened to lunch with 
me, he said to me that they were very good, " but, don't you tliink, 
II:iiiiil1oii, that they are. a little warm ?" "Warm?" I replied, 

mr*e they are warm; would you have had the Council send 
us down a cold set of resolutions on such a question ? " 



A 1 1 \ r . 58 . ] Excu rsion in Wick low. 1 6 3 



CHAPTER XLIX. 

EXCURSION IN WICKLOW. DR. INGLEBY AND METAPHYSICS. 
HAMILTON'S POETRY. * DECLARATION OF MEN OF SCIENCK.' 

(1864.) 

THIS year was with Hamilton one of enormous diligence in 
mathematical investigation and correspondence, his manuscript 
books and letters several times recording work extending in the 
day to beyond twelve consecutive hours.* The only intermission 
of any length which he allowed himself was occupied by a few 
days' excursion in July, taken in company with his daughter, 
through part of his favourite recreation ground, the county of 
Wicklow. His daughter notes that in this excursion, while 
evident tokens appeared of the decline of his muscular strength, 
his bright companionableness (shown, for example, by discussions 
with an intelligent Protestant carman on the parable of the Unjust 
Steward) and his youthful spirit of enjoyment were quite unim- 
paired. His own record of this excursion in a letter to De Morgan, 
dated New Hotel, Grlendalough, July 25, 1864, makes mention of 
an interesting fact of which I do not remember elsewhere to have 
met any trace, namely, his accompanying "Wordsworth to Glenda- 
lough. 

4 ... I had visited these lakes of Grlendalough in company 
with the poet Wordsworth so long ago as 1829, on which 

.ision I was young enough to climb from the lake-shore into 
St. Kevin's Bed an ascent not quite without dauger, at least 

* One letter to liis son Archibald says (June 6) : { This morning I was up at 
four A.M. ... 1 have been busy ever since.' Another (September 12) : ' 1 have 
been up since long before daylight this morning, but am not yet 

M 2 



1 64 Life of Sir William Rowan Hamilton. [1864. 

to people who have not learned to swim. But at the time I was 
familiar with both swimming and climbing. I have been in this 
neighbourhood many times since, but do not remember that I ever 
enjoyed more thoroughly a visit to it, or to various other parts of 
the Co. "Wicklow, than on this last occasion, when I have been 
touring with my daughter onhj : Lady Hamilton, though wonder- 
fully recovered from a recent illness, not feeling herself quite 
strong enough for the exertion and fatigue which last was not 
altogether to be despised by ourselves, especially as we chose to- 
travel back to back on an outside car. . . .' 

I select from his scientific correspondence a few passages which 
I think will be judged to be of interest either as throwing light 
upon his work and character, or as bearing his testimony to the 
qualities of his correspondents. 

From A. S. HART, LL.D., S.F.T.C.D., to SIR W. E. HAMILTON. 

1 TRINITY COLLEGE, January 25, 1864. 

' On Dr. Lloyd's return from Italy he told me of a conversation 
he had had with Plana about the quantity of heat received directly 
from the sun during the year at different points on the earth's 
surface. Not knowing where to find Plana's investigations, I 
attempted the calculation myself, but have got no results beyond 
those on the enclosed Paper, and they are so unlike what I had 
expected that my faith in them is weak, although I cannot detect 
any error. Did you ever happen to turn your attention to the- 
matter, or have you any Paper of Plana's on the subject ? ' 

From SIR W. E. HAMILTON to A. S. HART, LL.D., S.F.T.C.D. 

'OBSERVATORY, January 26, 1864. 

4 You have so remarkable a talent for simplifying things which 
appear difficult in Mathematics, that I by no means despair <>l 
following your analysis throughout in your last note to rn\ 
although I am rather rusty in elliptic integrals, with which, how- 
ever, I long ago connected (without publishing the connexion) the 
quaternions, as I had the pleasure of proving to my affectionate 
host of the time, Dean Peacock, at Ely, in 1846, when he asked 



M -.TAT. r*8.] Statics and Dynamics. 165 

me wliether I had yet thought of establishing such a connexion ; 
un<l I produced on the spot, from my modest luggage, a manuscript 
book containing a sketch of such an application.' 

Subsequently, * I have just an impression that Plana claims for 
Poisson the merit of having converted a discontinuous function into 
one of continuous form, ami asserts that Fourier (which, if true, is 
wonderful) had/at&rf, or at least omitted, to do this.' 

His next letter to Dr. Hart puts forward a notion illustrative 

of his characteristic power of generalisation; and his inability to be 
satisfied with superficial distinctions, however generally received. 
Its importance induces me to print it entire. 

From the SAME to the SAME. 

' OBSERVATORY, April 13, 1864. 

4 1 write just a line to warn you that I have indulged myself 
by taking a large ruled sheet to write to you upon, which may 
have a successor, but which you will be most welcome to throw 
-aside, whenever you may receive it, although any remarks of yours 
on the subject will be valued. Quaternions do not of necesxit>/ 
enter into the exposition of any part of the views on which I 
should thus like to consult you; but it is natural that they should 
have modified (ho formation of those views, so far as they are 
mathematical. 

' The main point is, that the whole of what we call STATICS is 
really a part of Dynamics, except in books ; there being no sucli 
thing as Rest in the world, unless (by a bare possibility) in the 
centre of gravity of the universe. 

' All this has a bearing on Foucault's Pendulum experiment, of 
which I think that I have lately given a very simple and satisfactory 
(but as yet unprinted) explanation by the Quaternion Analysis. 
.But I carry it severely on to what seems an incomparably simpler 
question, namely, to what I presume to call for the moment The 
Plumb Line Experiment ; tried every day by Mr. Thompson here, 
or by myself, when we close the shutters to let the plumb line hang 
undisturbed by currents of air, and see that the wire bisects a dot, 
-and continues to do so. 

* When that interesting event comes to pass, he says, and so do 



1 66 Life of Sir William Rowan Ha mil ton. [l w 

I, for ordinary purposes, that lite j>/innJ> line is at rest. But we do 
not need to be told nor does he that in point of fact, annual 
motion being abstracted from, the wriijht /.s in motion, describing (in 
this latitude) something like 900 feet eastward per second. Instead, 
then, of saying " What keeps the plumb line at rest ? " we ought, 
for greater accuracy, to inquire " What makes the weight describe 
its (tiit ma? orbit (the parallel of latitude) ? Of course, you see, the 
moment the question is stated, that the vis acceleratrix in this 
circular motion is the resultant of the earth's attraction, and of the 
tension of the string, at least if the mass of the suspended body be 
unity ; but is this at all a usual view ? 

' If you give me any encouragement I shall perhaps pursue 
the subject in letters to you, for a short time, in the least borixlt way 
of writing, which may be the most openly and ostentatiously borixh, 
by writing on large ruled sheets. But it is just as likely that, after 
indulging myself by expressing my views, I may quietly put the 
sheets in the fire/ 

In the month of May, Dr. Hart, who had been furnished by 
Hamilton with proofs of the Elements, now reaching to page 696, 
adds to an acknowledgment the following : * If I might venture a 
suggestion, would it not be advisable to close this volume with the 
geometrical applications and leave physics for a second volume ? ' 
In November his desire, on behalf of himself 'and the present 
generation ' to see the great work given, even in part, to tin- 
public having become in the meantime, very naturally, rather 
impatient, Dr. Hart repeats his suggestion, urging in support 
some very good reasons. Hamilton's reply seems to have 1> m 
only the sending to his friend some more proofs, which brought 
liim the gratifying reply which I here give : 

From A. S. HART, LL.D., S.F.T.C.D., to SIR W. E. HAMILTON. 

' November 23, 18(51. 

'I hasten to retract any objection I made to inclining Physi- 
cal Applications in tho Element* at least so far as Statics. 1 had 
no idea that you could condense so much into three ]>a^-. --. and I 
should be very sorry indeed that such a full and concise theory of 



\rr.\T.59.] Cost of Printing* 167 

the Statics of a Eigid Body should have been suppressed, espM-i;illy 
as the two ideas of the centre of a general system of forces, ami 
the tension of a system of equilibrating forces are new to me and 
perhaps to others also. 

'If your dynamical applications are equally concise and preg- 
nant, they will certainly form an admirable conclusion to the 
\V,,rk.' 

Hamilton was now again becoming anxious about the cost of 
printing and publishing his book. He shrunk from applying to 
the Board of Trinity College for more pecuniary assistance, and, 
in prospect of negotiating with publishers, thought that attestation 
from some high quarter to the value of his researches might aid 
him in gaining favourable terms. With this view he wrote as 
follows to his old friend Dr. Eomney Eobinson : 

From SIR W. E. HAMILTON to the REV. T. E. EOBINSON, D.D. 

1 OBSERVATORY, December 5, 1864. 

* I am, as I hope, approximating to the Moment of Projection 
or, in plain English, drawing to the time of publication of my 
long and laborious work, entitled the Elements of Quaternions. 

' It is only, however, within the last two or three days that I 
have opened negotiations on the subject with my old friend George 
Smith (Hodges, Smith, & Co.), and I have not seen him since I 
wrote last week, partly because I am hampered by a very heavy 
cold. But it is arranged that I call upon him, if possible, on 
Wednesday next (the day after to-morrow). 

* The total expense of the work which is not yet quite ready for 
publication, though very nearly so will have amounted to about 
400, whereof the College will have kindly borne 200, that is 
about one half. The remainder, even if there should be no re- 
muneration, however small, to the author , may well deter a cautious 
publisher. 

' Quite lately Dr. Hart who, as Senior Fellow, Bursar, and 
Friend, has watched the entire progress of this last work of mine 
has been pleased to write to me the kind note of which I enclose 
a copy made by my son William Edwin. . . . Dr. Hart and Pro- 
fessor Tait have been in fact the only persons to whom I have sent 



1 68 Life of Sir William Rowan Hamilton. 

copies of the printed sheets regularly. But I indulged myself 
lately by sending to you five half-sheets of my Content*, and al>< 
the last printed half -sheet of the Elements themselves. And it 
seems not impossible that even those may have served as some 
unit i- rials for forming an opinion on your part. 

1 If you can honestly say and I know that you will not say 
it otherwise that your general impression is favourable to the 
work, and especially that you think it treats old and celebrated 
subjects of science from a new point of view, I shall be glad: Imt 
you see that to render your reply useful, it should be prompt. 

* For my own part, on reading over lately those Essays of mine 
in the Philosophical Transactions, which won for me, thirty years 
ago, the applauses of the whole scientific world so much so that I 
have been quite tired, though of course flattered, by meeting so 
many references since to " les equations Hamilton," &c. my feel- 
ing was that I was not ashamed of those old Memoirs; but that 
they belong to a past age of Analysis, so completely do Quater- 
nions appear to me to furnish an avaxaivtaaif of mathematical, and, 
through it, of physical science.' 

* December 6. I find that I omitted to enclose, in my packet 
of yester-evening, the copy of Dr. Hart's note, which I now send. 
Of course Dr. Hart could have no motive to flatter me, and the 
warmth of his expressions is to be traced to a sort of penitence for 
his having very strongly, though with all kindness, urged me 
previously to let the E 'foments appear, as a purely mathematical work, 
all physics being reserved for some future occasion. But I am too 
old to trust to the future. 

* We all enjoyed much a recent visit from Francis Edgeworth 
who permits me in conversation to call him "Frank" for to 
me there can be no second "Francis." 

* AVitli kind regards to Mrs. Eobinson and to Mrs. Stokes, 
when you can forward them, I am,' &c. 

Dr. Robinson's reply was promptly despatched : 
Fi-om REV. T. ROMNEY ROBINSON, D.D., to SIR W. R. HAMILTON. 



< . :. . \IOKY, ARMAGH, JJcnmber 7, 1S(M. 

' I am very glad to find that by the approaching publication <!' 
the Element* wo shall be put in possession of your latest develop- 



AI:T\T. 59.] Romney Robinsoifs Testimony. 169 

nu.'iit of this magnificent branch of Analysis. As far as I can 
judge from the proofs of the Content*, which you kindly sent, it 
will be of great use in making more generally accessible the won- 
ders and the wealth of the field which you have so happily opened. 
The Statical and Dynamical applications of it in 416 and 417 
juv almost startling from their brevity, yet power and extent. 

1 1 hope nothing may occur to delay the appearance of the book, 
and in this I am sure all my mathematical friends will heartily 
concur. 

* I fear the Eoyal Society is not allowed by its laws to give 
pecuniary assistance to publications with which it is not officially 
connected ; otherwise I think an application to it would be suc- 
cessful. 

* I congratulate you on this conclusion of your great work : 
not as a final -resting-place, however, for the field is infinite. 

* Ever yours/ 

From SIR W. E. HAMILTON to REV. T. EOMNEY ROBINSON, D.D. 
[FROM A DRAFT.] 

4 OBSERVATORY, December 19, 1864. 

* I regret that I have not yet thanked you for your kind letter 
of ten days ago. It was alike honourable to you and to myself. 
I have not yet attempted to make any use of it ; but have allowed 
two recent guests of mine to read it. ... 

* My last visitor has been the Eev. Eobert Perceval Graves, 
brother of Dean Graves, and a very old and dear friend of mine, 
who slept here last night and left me to-day. The room which he 
occupied, and which other guests have occupied lately, shall be 
very much at the service of Mrs. Eobinson and you, if you should 
pay me the compliment of spending a few days and nights here, 
when the weather becomes finer. I am, &c. 

* P. S. When I know more, I shall mention more, of the 
prospects of my book. Meanwhile, like Milton, I bate no jot of 
heart or hope except as regards money, which is the least impor- 
tant item in the matter.' 

In June of this year Hamilton gave himself a little diversion 
in the way of arithmetical calculation, connected with astronomy 



170 Life of Sir William Rowan Hamilton. [1864. 

and chronology, as recorded in the following letter to his second 
son: 

From SIR W. E. HAMILTON to KEV. ARCHIBALD H. HAMILTON. 

'OBSKEVATORY, June 27A, 1864. 

' I have recently felt a curiosity to calculate back to the date of 
the Hegira (or Hejira), from the statement in the first page of 
Thorn's Almanac, that the Mahomedan year 1281 begins on our 
Gth of June, 1864, and from my remembering that a Mahomedan 
year consists of twelve lunar months, whereof the mean value may 
be safely taken as 29-5306 mean solar days. (Laplace, indeed, 
assigns 29-530588716 days as the present value, perhaps for the 
commencement of this century, of the mean lunation, or mean 
xf/nodic revolution of the moon ; but besides that this value is very 
nearly = 29*5306, the month is sloicly diminishing, by the secular 
acceleration of the moon's motion.) Multiplying then 29*5306 by 
12, I got, first, 354*3672 days as the average length of the 
Mahomedan year; and then multiplying this by 1280, I found, 
very nearly, 453590 days, as the interval from the Hegira (or 
from the commencement of the Mahomedan year 1) to the 6th of 
the present month. Dividing 453590 by 365-25 (which latter is 
Hie average length in days of the Julian year) I got near/t/ the 
quotient 1242; whence it might be at once inferred that the flight 
of Mahomet took place in the year 622 (A. D.). But to fix the 
time in that year, at least nearly, it seemed convenient to proceed 
as follows. 1240 Julian years = 310 x 1461 days, = 452910 thv 
above calculated interval 680 days ; and our 6th of June last was 
in Russia only the 25th of May; the Hegira then preceded the 
25th of May, 624, by 731 days. Hence (if the foregoing data be 
admitted), the Hegira followed the 25th of May by 51 days: it 
fell then on the 15th of July in tin' year (>%2 A. D. 

* Such, at least, is the final result of the foregoing calculation ; 
but I placed no great reliance on its minute accuracy till I su 

,itli/ found Jlorschel's Astronomy in one of the scientific shelves 
in the Library, and had the satisfaction of seeing, in page ' 
of that work, the precise date, JULY 15 A.I). 622, assigni-d as that 
of tin- ll.jira (so spelt by Herschel).' 

'What set mo thinking of the date referred to was my 



. 59.] Date of tJie Ifcgira. 171 

ntly happening to take up Helen's copy of Tin- Ailrcntarcx <>/ 
tlic Ca/iph Jfftroun AIruxclii<(, recount al />// //// Ant '/mr of Mnnj 
J'otcell . . . London, 1855. Who the interesting authoress is we 
forget, if we ever knew. There is a not unpleasant air of oriental 
learning, got up no doubt for the occasion, but not unfair, since 
some references are given in the margin. But the very first mar- 
gin al note is as follows : " Of the Birth of Haroun the Just. 
Year of the Hegira 139, A.D. 761." Then come at intervals the 
dates : 



A.D. 770. 



A.D. 786. 



A.D. 807. 



Heg. 148. Heg. 164. Heg. 185. 

* The process of manufacture, you see, is very simple. Miss x y z 
simply Kiihtrm-tx 622 from A.D., in order to get her "Heg."! I 
saw that this could not be right, whatever the A.D. date of the 
Flight may have been. Her rule would have made this year 
= Heg. 1242 ; whereas it answers to part of 1280, and part of 
1281, of the Mahomedans. I could not at first find a single book 
of dates, and was glad to calculate for myself as above, and I am, 
dear Arch, your affectionate Father/ 

That this stern detector of miscalculations in chronology was 
not disposed to be a harsh critic of authoresses is shown by the 
following extract from a letter written not long before the 
above : 

' OBSERYATOBY, April 29, 1864. 

'DEAR Miss ALCOCK, As you and I are both friends of my 
daughter, I may perhaps be permitted to consider myself a& 
entitled to offer my thanks to you for the very great pleasure (1 
may well add instruction) which I have received from the perusal 
ot your beautiful and valuable little work on Alfred the Great 
and Good. It may not be known to you, that Mr. Wordsworth, 
the Poet, . . . claimed to be a lineal descendant of Alfred. . . .' 

For a considerable time in the course of this year Hamilton's 
mind turned to metaphysics; and Plato, Aristotle, Descartes. 
B.Tkeley, Kant, Coleridge, Sir W. Hamilton of Edinburgh, and 
Dr. C. M. Ingleby furnished him with matter of thought. This 



172 Life of Sir William Rowan Ham Uton . [ 1 864. 

digression from his ordinary occupation, or rather I should say 
these occasional diversions, were caused hy his having received 
from his new friend, the last-named in the above list, the First 
Part, recently published by him, of an lutrwliu'tion to Metaphysic. 
I prefix to the extracts from letters of this year to Dr. Ingleby 
one written in 1861 on the morning of the day of which the 
eveniug was spent by the two friends at the Observatory in their 
first conversation on Philosophy.* It fitly introduces the rest of 
the set by which the reader is prepared for still further treatment 
of the same subjects in letters to other correspondents. Whatever 
may be the inherent value of the contents of these letters and on 
this point I do not pretend to be qualified to pronounce an 
opinion they are characteristic of the writer, and deal with inte- 
resting topics. 

From SIR W. E. HAMILTON to C. M. INGLEBY, D.C.L. 

'OBSERVATORY, August 21, 1861. 

' From your extremely obliging note received on Monday 
morning, I collected that unless I wrote to the contrary, as 
being otherwise engaged, &c., you would favour me with a visit 
this evening. I pray you to accept that impression as an excuse 
for my not having written to you since, though I might mention 
that I was working, on Monday, for example, for more than ttrdrr 
consecutive hours, on things connected with my forthcoming 
volume, the Elements of Quaternions. I happened to notice this 
morning, while relaxing myself for change of labour is, to a 
>tudious man, a relaxation with some perusal of Plato, in a v;ir-l 
folio of M.DCII., that 9 Quaternities had been picked out of 
writings by old commentators, e. g. 

SECUNDA. 

Cratylus : vel do recta nominum ratione. 
Thcietetus : vel de scientia. 
Sophista : vel de ente. 
Ci villa: vel de regno. 

Xitjira, p. 135, where the Christian name of Dr. Ingleby is printnl 
'Charles' instead of 'C 



58.] Dr. In gleby and Metaphysics. 173 

* I rarely look at the Latin column ; Imt in this case there was 
no help for it, because there was no Greek to match. 

' If anything should unexpectedly prevent your coming here 
this evening, I beg you to believe that I have attained so far, in the 
hfudy of Plato, as to be most unfeignedly aware of my imni> 
inferiority, in Philosophy, to the Sir "William Hamilton.' 

From the SAME to the SAME. 

1 OBSERVATORY, May 26, 1864. 

* ... Your Part i. [of Dr. Ingleby's Introduction to Metaphysic] y 
above acknowledged as arriving, has interested me extremely. . . . 
It is long since I have read metaphysics, as in any sense a student ; 
but the Paper quite revived my old interest in such subjects. I 
felt that I examined better on the Tuesday in consequence of 
having so spent the evening previous ; and on the following 
Wednesday, when I went to dine with a co-examiner, Professor 
Jellett, at whose house (seven miles at least from this place, accord- 
ing to my estimation) I met, among other pleasant people, my old 
friend Dr. Lee, now Archdeacon of Dublin, I took Part i. with me 
to read in the cab, and to think on; and on returning to this 
Observatory, at about half-past twelve, sat up for at least two 
hours, to read and think on it again. 

' I had retained a recollection, confirmed by your letters, of 
your being a very agreeable and highly cultivated gentleman ; 
but was scarcely prepared for finding you turn out so much a power 
in the world as your new " Parts " seem likely to rank you. If 
you had published anything previously, I should be very glad to 
see it, and very willing to purchase it : for I like your style. " Le 
style, c'est 1'homme," say some: it gives at least an indication. At 
the same time, as a sermon which everybody praises does no good, 
you would not be satisfied, I suppose, if I had no objections, or at 
least difficulties, to offer for your consideration. 

' Thus in 6, sentence the second, I should like to know 
whether you deliberately teach, respecting the Ego, and the Non- 
Ego, " that whatever is not one is the other." If you do so teach, 
I still do not understand, as a logician, your saying immediately 
afterwards, " i.e. that they exclude each other." Surely A and B 
may exclude each other, without the non-existence of some third 



174 Life of Sir William Rowan Hamilton. [1864. 

thing, or thought, C. At this moment, and with a deep sense 
of my want of recent preparation for entering on any such high 
investigation, I should be disposed rather to form this little tabl.- 
or scheme (not strictly in the Kantian sense, perhaps, of this 
word) : 

( Der Ich. Das Nicht-ich. 

( Die Gremeinschaft. 

or in Latin : 

Ego. Non-Ego. 

Medium (Communitas) . 

'On turning to your 64, page 61, 1 see that you use the ironl*, 
" a medium between Ego and Non-Ego " ; but, as it seems, rather 
to reject a view in which " Organism " might be taken to be such. 
What you say in the same paragraph, but in the following page 
(62), comes much nearer to what I mean. You say : " How Ego 
can be disturbed by Non-Ego is, indeed, an inscrutable problem. 
The fact must be taken as the subject of a primitive faith." But 
to complete for the present purpose the outline of my view of th 
above hinted-at Medium transcendentale, between the Ego and the 
Non-Ego, I would refer to your 68, in which you say that 
" Correspondence and mutual dependence without likeness is the 
watchword of this system." And what is still more important for 
the immediate purpose, you say, in the same page, " every change 
which we voluntarily effect in the latter world " (that of the senses) 
"has its concomitant and corresponding change in the fornu-i 1 ." 
You admit, then, that the Ego can disturb the Non-Ego, although 
in a manner to us entirely unknown. This I call a part of 
MEDIUM (perhaps rather a phase of it. I am reminded of Coir- 
ridge's "Punotum Indifferens," sive " amphotericurn," which ;i> 
viewed from A is B, and as viewed from B is A) ; the other part * 
which we know, because we construct it t according to your very al>lc 
and important development of Kant's ^Esthetic, if rightly under- 
stood by me, is what we call Perception. In the long drive t 
JellottV dinner on Wednesday last I had the greatest comfort in 
>;i v ing to myself, Tlmt free, that spire, is truly t/tere, as I see it 
becaii>- 1 liuvr roiiMnirtrd it, though not without sugp->tion I'roiu 
tin- Non-Ego, ats an </////r/ so located. I had no difficulty in turn- 
ing to the passage from which your motto is takm, and which ver) 
btruck mo in tin- AY////,-. 



. 58.] Sight and Touch. 175 

k 1 Loping that you will not regard me as having been too free 
in my remarks, I am/ &c. 

OBSERVATORY, June 14, 1801. 

1 1 have no letter of yours, nor of my own, beside me at this 
moment; and there are at least two unsent letters of mine, to 
which you are entitled. When I have written a letter I am apt 
to think my duty to my correspondent performed ; the additional 
circumstances of folding it up in an envelope, directing, stamping, 
and in some cases copying, or getting copied, the contents, appear- 
ing to fade away into a remote perspective. . . . 

* You mentioned, I think, a commentator who conceived, ac- 
cording to you, that Aristotle had written in Latin. I rarely read 
new novels (though I read fast), but am never tired of Walter 
Scott. Well, I yesterday lit, in his Woodstock, on the passage 
where the learned Dr. Rochecliffe remarks to Joceline : " Where, 
says the Septuagint, Percussum Egyptium abscondit sabulo." 
This is only one place, out of several, which convince me that 
Scott actually believed the LXX. to have been written in Latin. 
Did he coin the quotation ? I have not a Vulgate at hand.'* 

* June 4, 1864. . . . You were pleased to express, in a recent 
note, a fear, or at least an expectation, that some people will 
blame, or "superciliously smile at" you, for the "constant intru- 
sion of physical and physiological considerations into Metaphysic." 
I shall not be one of those people. On the contrary, I conceive 
that it is most strictly proper and philosophical, in connexion with 
Theories of Perception, to try all possible experiments: and I may 
shortly indicate a certain " corpus," -videlicet my own, on whicli 
some such might perhaps usefully be made. (Lest I should not 
have time to return on this, let me just say that I allude to my 
habitually seeing double since I was very young. I attribute this 
chiefly to my having had a telescope of my own a very good 
sliding Dollond when I was only about eight years old. . . .) 

' You have most justly insisted on the immense superiority, 
which in fact I think cannot be exaggerated, of the Sense of Siyht 
above the sense or senses of Touch. To hear some people talk, 
even people of highest note, or rather to read their writing 

* It is in the Vulgate, Ex^d. ii. 12. 



176 Life of Sir William Rowan Hamilton. [1864. 

would be apt to suppose that a man blind from birth should actually 
li.-ivo an adrantagc over a seeing one, in learning such a science as 
Geometry ; that all would come riyht to 7//w, almost from the/?/v^: 
or at the very least, that he would have/6W<?r illusions to unlearn. 
Yet the existence of any such natural superiority of the born blind 
does not seem to be confirmed by experience. (Please to observe 
that I do not at all dispute the possibility of a man born blind 
attaining any height in mathematics : I only deny that he has any 
advantage over seeing people. Sanderson was certainly a remark- 
able man of science ; but he had at least a year of vision.) For 
my part I have very long been of opinion that Touch even 
muscular touch, handling as distinguished from mere feeling has 
no advantage whatever over sight even over vision with a SINGLE 
EYE and this is a very important assertion for my purpose, in 
making known, or even suggesting to us the Existence of Objects, in 
TRIDIMENSIONAL SPACE. 

' As to mere skin-touch, such as that with the palm of the hand, 
or other soft part of the body, the information of objects which it 
gives seems to be almost nothing, even at first contact : and in an 
extremely short time, such information, or even suggestion, appears 
to fade away completely/' 

* June 4. ... I doubt whether skin-touch would have led us to 
a belief in objects at all, any more than the sense of smell : less 
easily, I think, than that of taste, on account of the movements of 
the tongue ; but I admit it to be impossible to say, what the slightest 
.N/A/y/r might or might not have done, when falling from what we 
call without, on the powder magazine of our Potential Intuition. 

* But take our Muscular Touch HANDLING. This sense gives, 
I think, Two Dimensions of Space, with ease; but not with greater 
ease (nor with so much to me, who am not blind) than the sight of 
a SINGLE eye, with its power of LOOKING, first at one part, then at 
aunt In r j>art of the Picture. Books tell me you tell me that in 

looking I alter the direction of an Oy>//> A.ris ; and I am far 
from disputing your authority. I submit to the teaching of 
OI-IK \\. ANATOMY. But MUSCULAR ANATOMY has other revela- 
tions to make, or doctrines to propound, which I am still less 
qualified to dispute ; and of which I need only say that they 
intio'luco the element of Body our own body at least as much 
into what may seem the province of WILL, in the Act of handling, 
as in that of looking. 



A i :r AT. 58.] Berkeley. Coleridge. Sir W. Hamilton. 177 

' To get 'beyond cither, something deeper, higher, more powerful, 
is required, namely, the Original Intuition of Space, which waits 
to be awakened by Sensation* 

1 June 15, 1864. After laying the two preceding sheets aside 
for more than ten days, I am astounded, or at least amused, at my 
own audacity, in having written here and there, as if I had a right 
to Jin opinion of my own on such high points of philosophy. And 
you may be amused or surprised when I tell you that I have not,. 
at present, a single book of Berkeley's or of Hamilton's in my 
library. 

* As to Berkeley, it is of the less consequence, because I was early 
lent a good three- volume edition of his works by a noble friend, 
who was formerly a pupil of mine, and who, after about twenty- 
five years, reclaimed the loan not very long ago ; so that I had 
leisure to become sufficiently impregnated with Berkeley's teach- 
ing, for one who has never aspired to be himself a teacher of Philo- 
sophy. In fact, when I was rather young, namely, in 1832, I 
allowed Coleridge (at Highgate) to see that I was at that time 
a regular Berkeleyan ; and he was pleased to say for our 
several interviews, of that year and the following, of which some 
were long, were not all monologues on his part he allowed me to 
make a remark now and then, and actually modified his discourse 
to meet it: "Oh, sir, you will grow out of that! " In some respects 
that prophecy has been since fulfilled ; but out of love and reverence 
for the great and good Bishop, I trust that I shall never grow. 

Since, then, I have sat, although long ago, at the feet of 
Berkeley, what more disqualifies me than the temporary absence 
of his Works from my library, to be a worthy correspondent of 
yours, is the extreme slightness of my acquaintance with the 
writings of the great Sir William Hamilton in whom you seem 
to have formerly recognised a master. (I have, however, read,, 
several times over, the "Bampton Lectures" (for 1858) by Dr. 
Mansel, one of the most able and affectionate pupils of Hamilton,. 
with its Preface, and most of its Notes [3rd Edition] . . .) A 
young friend lent me, in 1850, "Hamilton's Eeid": this is 
probably not the exact title of the book, which he left with me for 
at least a year, perhaps for two. I was at that time very busy in 
carrying through the press my Lectures on Quaternions : as the 
Table of Contents of the Elements a quite new work on the same 

VOL. III. N 



178 Life of Sir William Rowan Hamilton. [1864. 

general subject is occupying me just now. But I have a lively 
recollection of running over, as I might have done a clever and 
spicy Article in some Review, an APPENDIX such, I think, was 
its title in which Sir William Hamilton had (to me at least, and 
at the time) the air of vindicating what he was pleased to call the 
Doctrine (or something equivalent) of COMMON SENSE, by a tre- 
mendous artillery of learning and quotations ... On the whole, 
I very much enjoyed the Appendix; but rose from it with the 
feeling that no conviction had been imparted or disturbed.* 

' I totally forget if I ever took the trouble to understand what 
" cosmothetic realism" and "cosmothetic idealism" may have 
meant in Sir William Hamilton's Vocabulary. Now your 
pamphlet has set me thinking, with some curiosity, on such things 
again ; but it has naturally produced an impression not painful, 
because I am otherwise well employed, and there must be a division 
of labour that I do not know the A B C of your present contest 
with your former master. 

' . . . But, talking of instinct, it appears to me that no theory of 
Human Perception can be completely satisfactory which ignores 
Animal Perception. In Campbell's Specimens of the British 
Poets I have just noticed the following lines of an obscure writer 
(William Whitehead) : 

"The kitten, too, was comical, 
She played so oddly with her tail, 
Or in the glass was pleased to find 
Another cat, and peeped behind." 

Whether this be true of a young kitten, I doubt : the lines perhaps 
do too much honour to its intellect, or imagination, or even percep- 
tive power. But I have frequently seen a wise old cat look at her- 
self in a mirror, and then, at once, go behind it. Have you ever 
troubled yourself to speculate upon this fact? And generally 
should you admit or deny that there is a good deal of instinct in ' 
our perception ? 



* On the 22nd of September, 1864, Hamilton writes to De Morgan : * I con- 
tinue to admire Hamilton himself [Sir W. H. of Edinburgh], as a writer, and 
should gladly be a student of his works, for many years to come, but am satis- 
fied that I should never become & pupil. His whole view of what l'n 1 1 .<<n -n v 
it, or ought to be, differs essentially from mine. But it would not become me 
to enlarge at present on such a topic . . .' 



A i :T \ i . 58. ] The Stereoscope. 1 7 9 

' Human babies, I suppose, do not really see, until they begin 
to take notice, however good the optical imnyrs upon their retinas 
may be. But as to those eye-images, true as it may be that in 
strictness we do not see them, and even that, as being thus unper- 
ceived, they do not (Berkeleyanly) exist, metaphysicians ought not 
to forget that physician* hold them to be, at least under ordinary 
circumstances, conditions of vision. 

1 1 am pretty sure that Sir Isaac Newton has somewhere said 
that the great object to be aimed at, in the improvement of optical 
instruments, is to make such (retinal) pictures "bigger, brighter, 
and more distinct": or words k> that effect. 

* This has been so rambling a letter, if a letter it may be called, 
that I shall follow up here a remark on its 2nd page, by observing 
that although I habitually see a double universe, yet a marked im- 
provement has taken place within the last few weeks, in my power 
of seeing single. This I attribute to my having lately, for the first 
time in my life, bought a stereoscope, and used it at leisure here : 
no doubt incited by your " Part." A friend, within a few minutes' 
walk of me, has long had a stereoscope apparatus ; but years elapsed 
before I could catch the effect at all. With each eye, separately, I 
saw a good relief; but it was at Cheltenham, in 1856, that I first 
was able to see that TERTIUM QUID, which is the true result of the 
stereoscope : and certainly it greatly astonished me. The illusion 
was wonderful ; but I cannot yet feel that it throws much light on 
my own ordinary process of vision. "Whatever Berkeley may have 
said, I am quite sure that / SEE DISTANCE with each eye separately ; 
and although the two focal lengths are not exactly equal, the uni- 
verse seen with the one eye differs in no appreciable degree from 
that perceived with the other. 

*I entirely repudiate the notion that it is tangible distance 
which I see : indeed I am not sure that there is any such thing as 
" tangible distance." What I see is, that some things are nearer 
than others ; that there is a gradation, a perspective, which is almost 
exactly the same to one eye as to the other. It is true that, //' 
My attention is called to the question, when I look (for example) 
from a window of an upper room on trees or shrubs in a lawn, I 
can con a- ice- myself going to the room-door, thence to a lobby, 
thence down the stairs, thence out at a hall-door, and thence 
walking across the lawn, till I put my hand upon a tree or shrub. 

N2 



180 Life of Sir William Rowan Ham ilton . [1864. 

But such is not my habit ; nor do I admit that any such reference 
to a possible touching forms even an element in the visual perception 
of distance , which I insist that I HAVE (and with each eye), whether 
I ought to have it or not. As to how far off' any given object of 
vision is, I admit that my estimate is very vague : or rather, I 
make habitually NO such estimate at all, as referred to feet, yards, 
or miles. I only judge, or rather SEE, as above said, that one 
object is nearer than another; and this often with an extreme 
variety, as in the case (for instance) of a landscape. 

1 The result of all this is, that /cannot attach so much importance 
as you do to binocular vision. If it had pleased Providence to 
bestow on me a second pair of eyes, I trust that I should have 
borne the infliction with patience, and have learned to believe in a 
single universe, while perhaps habitually seeing four. 

* Having been from childhood a reader, and even to some 
extent a writer, if childish journals intermixed with scraps of 
Persian and Sanscrit, &c., are to be accounted writings, I certainly 
see a page of a book, or a sheet of a letter, as single. And when I 
succeed in so seeing a somewhat more distant object as at this 
moment a cup of tea upon a table not very near me I am con- 
scious of some new satisfaction, and must admit that, in a way 
which I cannot thoroughly explain, I have more than before the 
sensation or impression of reality. This result may be partly physi- 
cal, the object being then more distinctly seen ; and perhaps partly 
mental, inasmuch as I had not previously believed in the two cups 
which I saw. I am almost inclined to admit that, as with the- 
stereoscope, but in a much fainter degree, I see a tertium quid. But 
I am not conscious and have tried the experiment very often of 
any change of visible distance, such as I constantly, or at least 
normally, experience with the stereoscope, when I succeed with it, 
after some short double vision. 

' Some people would not choose to be so candid . . . and I am 
not aware that any one could guess the fact [of my double vision], 
if not informed of it by myself. I had the honour, forty years 
ago at which time I was a boy of being introduced to Miss 
Edgeworth at Edgeworthstown ; and she was pleased to indulge 
me with what I may venture to call an intimate acquaintance, or 
rather friendship, from that time to her death. She was an emi- 
nently truthful person, and it occurred to me to ask her, very early, 



AETAT. 58.] Binocular and Unocular Vision. 181 

when we were by ourselves, whether I squinted. She looked 
steadily at my eyes for a minute or more, and then replied : " No, 
Mr. Hamilton, you don't squint." On the other hand, I once 
knew a very amiable lady, who squinted awfully; it was really 
painful to look at her face : and yet I have no reason to suppose 
that she. saw double* 

1 The case may be more common than it is known to be ; as 
" Daltonism" was found to be not peculiar to Dalton, who told me 
himself, that he could not distinguish by sight, or rather by colour, 
the fruit from the leaves of a cherry-tree. I am not aware of any 
deficiency of that sort in myself. 

' To wind up the statement of my own belief, founded partly 
on experience, and partly on theory, regarding my own visual per- 
ceptions, I maintain, or at least believe, that if I had been blind of 
one eye from my birth, and had also, in some way, by malformation 
or paralysis, been deprived of the use of both hands from birth or 
earliest infancy, ... I should still, by LOOKING, and LOCOMOTION, 
have learned to SEE, with the ONE sound eye, precisely the SAME 
VISIBLE UNIVERSE, including gradation of distance (I do not say 
estimate of distance, which, as already remarked, I do not habitually 
make), as that which, with that eye, I now behold. I remain,' &c. 

'OBSERVATORY, June 21, 1864. 

* I received your letter of the 20th to-day but for the moment 
can only acknowledge it, by getting stitched into this sheet a 
number of others which had been lying by me [viz. the letters 
above printed from June 4th onward], and indeed are little worthy 
to be forwarded. Let me, however, say that I think I understand 
your Law of Reciprocal Causation (p. 61, &c.),t and have great 

* [Dr. Ingleby annotates " I have. C. M. J."] 

t ' . . . The principle of Reciprocal Causation is too much an individual 
insight, of power and vitality, for the intrusion of Scepticism. In the table on 
page 64 I might, perhaps, have preferably employed perceptus instead of per- 
eeptio. But in the singular it is an insolens verbum. If a reader can securely 
catch the great distinction of subjective and objective experience (i.e. qua per- 
cipient, and qua speculator), he will not be long in perceiving, as I do, that the 
Understanding (Verstand) erects the phantasmata ccsthetica into an objective 
mundus sensibilis ; and resolves the Real Mundus Transccndentalis into a 
Percept. Obviously to do these two things is, in effect, to convert the order of 
causation. Who has ever seen, or, at least, said this before me ?' [Extract from 
Dr. Ingleby's letter of June 20, 1864.] 



1 82 Life of Sir William Rowan Hamilton. [1864. 

hopes that it is both new and true. It certainly has given me 
several subjects of interesting meditation ; but to render an opinion 
of mine of any value whatever, on such a question, I ought to have 
been more recently reading Kant and other metaphysical writers 
on the subject of PERCEPTION : and, above all, since you so often 
refer to him, and in a controversial way, I ought to have made 
myself acquainted with the views of Sir William Hamilton, other- 
wise than by an imperfect recollection, referred to in sheet 4 of the 
enclosed letter, of a partial perusal of his work on Eeid, which was 
returned by me to a friend about twelve years ago. (I have lately 
been reading, with much interest, a good deal of Eeid himself.) 
P.S. Descartes, in Cap. vi., xx. of his Dioptrics, says . . . " Ex 
quibus fit, ut ne quidem sensus noster communis, ideam distantij 
capere posse videatur, ultra centum aut ducentos pedes abductae. 
Atque hoc patet ex eo quod Luna, et Sol, qui sunt e numero cor- 
porum remotissimorum quse contueamur, . . . pedales ut plurimum, 
vel ad summum bipedales nobis videantur, licet ratio dictet, illos 
longe maximos et remotissimos esse." Berkeley, then, was not the 
first to think, with his Hylas, that the moon appeared to be "about 
afoot in diameter." And some such estimate I believe to be very 
common, although I was too early spoiled by science to have a 
recollection of my ever thinking so. A foot, however, rather 
exceeds the average estimation, so far as I have noticed. Just 
now, I asked my daughter, " How large is the moon in the sky ?" 
"About half a degree," she replied. "Come," said I, "you 
learned that from Astronomy ; but answer as a girl of common 
sense." She knew, unluckily, that all depended on the supposed 
distance; still, by way of saying something, she named a small 
saucer as a comparison. I knew, long ago, a very short-sighted 
gentleman who, perhaps in consequence of the great irradiation in 
his eyes, compared the visible moon to a small round fable, which 
might be of the extreme bipedal standard of Descartes. People in 
general, I fancy, don't trouble themselves to estimate the distance 
of the moon, as seen by them, but think that they can estimate ite 
visible size, in feet or inches, or by some comparison of that sort.' 

The following letter to Mr. Barlow, F.T.C.D., on Sir William 
Hamilton of Edinburgh, on ' Hamilton's Reid,' and on Descartes, 
is in natural connexion with the above Ingleby correspondence, 



AETAT. 58.] Sir W. Hamilton. 183 

by which Hamilton's mind was led into the region of thought 
which in this letter he continues to explore : 

From SIR W. B. HAMILTON to BEV. J. W. BARLOW, F.T.C.D. 

' OBSERVATORY, June 30, 1864. 

* ... I have already been reading a good deal of the book you 
re-lent me yesterday (for this current month), known, I suppose, as 
" Hamilton's Beid." Beid's own works (at least his Essays) I had 
lately been reading with interest, in a well-bound two-volume 
octavo edition (Dublin, 1786), which was given as a College prize 
to the deceased uncle (Bev. James Hamilton of Trim) who edu- 
cated me. 

' Of Sir William Hamilton I am prepared to think even more 
highly than I already do. Were you not (I think you were) at, at 
least, one of the tea parties in Edinburgh in 1850, at Sir William 
Hamilton's house, to which we were invited together ? 

* I knew that he was an tfM^'-mathematician ; and had made up 
my mind that I would submit without reply to any attack on 
mathematics, from one who was so much older and more celebrated 
than myself, and was in infirm health besides. But Sir W. H. 
was so entirely the gentleman, as not to put my forbearance to any 
such test ; and I still preserve and value the rather rare book, of 
somewhat modern Latin poetry, in which Leibnitz figures exten- 
sively, and which he was so good as to request his Lady Hamilton 
to fetch down for me from a shelf known to her. 

' Well, I retain all my old respect for Sir William Hamilton, 
of Edinburgh, as a gentleman, and also as one of the most learned 
metaphysicians of modern times, at least within these countries. 
As regards the logical controversy between him and De Morgan, 
about the quantification of the predicate, and all that, I do not presume 
to have even an opinion, at present : perhaps, two or three years 
hence, I may be led to form one. (At Oxford, in 1847, I remem- 
bered enough of the, then, recent controversy, to set Mr. Hallam 
(of the Middle Ages, &c.) thinking from one breakfast to another. 
At the second breakfast Mr. Hallam admitted that the point was 
made out. By the point, I mean De Morgan'* view, as stated by 
myself.)* 

* See vol. ii., p. 495. 



1 84 Life of Sir William Rowan Hamilton. [1864. 

* But I conceive my old mathematical crony and correspondent, 
Professor De Morgan, to have proved in the Athennum, a few 
years ago that Sir William Hamilton had never attained to a 
correct understanding of the First Proposition of the First Book of 
Euclid; although he used to lecture thereon, for the purpose (I 
believe) among others of illustrating his thesis (note to Reid, 
p. 709), that "mathematics are, as is universally confessed, the 
easiest of all sciences. 19 The italics are his own). This seems to 
me unpardonable, in him. I am most willing to learn from, but 
how can I trust a teacher, who ventured to pronounce on the intel- 
lectual tendencies and position of a science, in which he was 
ignorant of his own ignorance of the first rudiments? Still I am 
very glad to have his book.' 

* July 4, 1864. Since my letter of Thursday last has not yet 
been posted, I wish to write a sort of postcript to it. ... 

'I have an Elzevir edition (Amsterdam, 1677) of " Renati 
Des Cartes Principia Philosophise." It is said in the title-page to 
be : " Ultima Editio cum optima collata, diligenter recognita, et 
mendis expurgata." It contains several distinct treatises, bound up 
together, e.g. " Dissertatio de Methodo," " Dioptrice," " Meteora," 
&c. But the weightiest is the Principia; a title and (for the 
age) a success which perhaps, nay, I think probably, stimulated 
Newton to produce his own great work, although it was Principia 
Mathematica. 

* As early as page 2, Des Cartes introduces his celebrated 
" Cogito " argument, under the form : " Ac proinde hoec cognitio, 
ego cogito, ergo sum, est omnium prima et certissima, quse cuilibet 
ordine philosophanti occurrat." Observe the ego : it is repeated in 
page 3, where he does not deny ..." non ideo negavi, quin ante 
ipsam scire oporteat quid sit cogitatio, quid existentia, quid certi- 
tudo," &c., but maintains that philosophers have rendered things 
obscure, which were in themselves most simple and well kuown, 
through seeking to explain them by logical definitions. 

* Now this " ego," which Des Cartes thus repeatedly couples 
with "cogito," has very commonly been omitted in quotations; 
and I have even seen arguments founded on such omission, which 
sought to prove that the author of the sentence had designedly kept 
the pronoun out of view, and was thereby .wp/tixficttl. His own 
remarks upon the whole subject are at least very clear. 



AETAT. 58.] i Ego cogito, ergo suvi? 185 

' My edition of Des Cartes' philosophy, as already mentioned, 
is of 1677 ; but it was not the first, and Spinoza had previously 
written a sort of mathematical commentary on the first two Books of 
that " Principia." Accordingly, I have an octavo volume of 700 
pages, published also at Amsterdam, but in 1663, and therefore 
14 years earlier, of which a part of the title-page is as follows : 
"Renati Des Cartes Principiorum Philosophise Pars i. et n., more 
geometrico demon strata) per Benedictum de Spinoza Amstelodam- 
ensem. Accesserunt ejusdem Cogitata Metaphysica." . . . The 
volume contains also a " Tractatus Theologico-Politicus " . . . 
(Hamburgh, 1670), from which Colenso may have taken hints: 
besides Epistles, &c. Spinoza, in his page 4, uses twice, without 
pronouns, the formula : dubito, coyito, ergo sum. 

* ... There seems to be an incompleteness in the copy of the 
Notes on Reid which you kindly lent me. It breaks off abruptly 
at page 914, in the middle of Note D * * * ; after which page 
come a couple of specimens of works preparing for publication. 
Have you a second volume? From the "Advertisement" I 
suspect that the volume, as published, may have abruptly ended ; 
but in the body of the work I have observed references to Notes 
G and R, for example (compare the following passage), which are 
either non-existent, or at least non-apparent. 

* In particular, and as connected with part of this letter, I 
observe that in p. 268 b, Reid says of Des Cartes : " He used this 
argument, therefore, to prove his own existence, Cogito, ergo sum." 
But in a footnote to the end of the following sentence (p. 226 a), 
Hamilton remarks : " On the Cartesian doubt, see Note R. H" 
Now, as above said, I can find no note later than D * * * ; nor is 
even that note finished in your copy. But perhaps I shall find 
the cream of what Hamilton had designed to say in some of his 
foot-notes to Reid. 

' I see that I must read Note C before Note A, on account of its 
technical terms.' * 

In a very interesting letter addressed by Hamilton to Dr. Hart 

* In Mr. Barlow's reply he says: ( " Hamilton's Reid" was, I believe, 
never completed, which is the more extraordinary, as Sir W. H. undertook and 
completed the editing of all Dugald Stewart's Works subsequently to the pub- 
lication of the edition of Keid.' 



1 86 Life of Sir William Rowan Hamilton. [1864, 

on the 30th of June in this year, he thus refers to the metaphysical 
excursions, a record of which has just been placed before the 
reader : 

4 ... The fact is, that one of my early tastes was for meta- 
physics, and something has lately occurred to revive it. Another 
was for Eastern languages ; and I chanced yesterday to light on 
the first sheet of a " Persian Grammar," written by myself forty 

years ago. These things, with others, may occasionally relax the 
Low: kk non semper tendit . . ." ; but "many tastes, one power" 
and my only power is mathematics.' 



This verdict, so deliberately pronounced by Hamilton, near 
the close of his life, upon his special faculty and function in 
relation to other constituent elements of his intellectual being, is- 
in remarkable accord with the expression used by him at the com- 
mencement of his public career, when, in a letter to his sister 
(vol. i., p. 286), he spoke of his interest in Science as being his 
master-passion. Doubtless the word ' others ' of the clause, ' These 
things with others? in the passage above quoted, was intended to 
include Poetry : and the passage is therefore scarcely compatible 
with a declaration which has been attributed to him, * I live by 
mathematics ; but I am a poet.' I find it difficult to suppose this 
to be an accurate report of his words. He might very conceivably 
have said, ' What / am as a man is more shown by my poetry, 
which reveals the inner current of my life and my affections, than 
by my mathematical works ; as also by my metaphysical, ethical,, 
and religious opinions, which indicate my standpoint in Philo- 
sophy ' : this it would have been natural for him to say, for he felt 
it acutely,* but we may consider as certain that he would not, on 

A letter to myself gives expression to Hamilton's painful sense of the 
inferiority of mathematics, exemplified by the works of the highest mathe- 
matical genius, to corresponding works of poetical genius, through the absence 
from them of elements immediately affecting the spirit and life of man. 
t< htimony borne in the first portion of the letter to Wordsworth's early appiv- 
ciation of Tennyson had been anticipated by the elder poet's written words ia 



AETAT. 58.] Hamilton s Poetry. 187 

the ground of the occasional expression of such opinions in letters 
and lectures on Astronomy or of the occasional relief of his feel- 
ings afforded by his poetical compositions, have claimed a place in 
the rank of acknowledged Philosophers or Poets. He may, indeed, 
have heen conscious of being potentially a Poet and a Philosopher, 
and that, I think, is all that he can ever have intended to express ; 
but however largely he may have felt that he possessed the ele- 
ments which constitute either the Poet or the Philosopher, it is 
satisfactory to read his clear recognition that his chief faculty was 
the power of dealing with mathematical truth, and to extend its 
boundaries his highest function. What he considered the main 
value of his poetry has already been recorded in the passages to 
be found in the second volume of this work, pp. 402, 612, 613 ; 
but I am able to add another more definitely acknowledging 
his sense of its defects, while referring to a favourable judgment 
on it of Francis Edgeworth, in which he evidently concurs. 

one of his letters to Hamilton (vol. i., p. 403), a fact which Hamilton seems to 
have forgotten, and, with regard to his notion of sending to Tennyson in return 
for pleasure and benefit derived from his poetry a copy of the Lectures on 
Quaternions, I remember discouraging it, on the ground that the poet could 
receive no reciprocal enjoyment from the book, which, in my opinion, was of too 
great importance to be used as a mere token of homage and obligation. It may 
be doubted whether my advice, which Hamilton acted on, was, taking all things 
into account, judicious ; but its motive was what I have stated, and if it was 
mistaken, I have only to hope for the pardon of the Poet Laureate. 

From SIR W. R. HAMILTON to R. P. GRAVES. 

* OBSERVATORY, July 18, 1855. 

' MY DEAR ROBERT GRAVES, I have many social and affectionate debts to 
pay you, but it suddenly presses on my mind, that although (thank God) I am 
in excellent health just now, yet I ought to put beyond the chances of mortality 
my written testimony to the discriminating generosity of our great and departed 
friend, the Poet Wordsworth. He has been thought by some to have been un- 
willing to allow praise to other poets and I never could enter, nor pretended to- 
enter, into his criticism on Burns 's "Scots wha hae wi' Wallace bled" but 
you know that Wordsworth admired and loved Burns, and could appreciate him, 
and deeply and long regretted that he had not been acquainted with him. Mr. 
Wordsworth (we talk still of Mr. Pope) gave me a copy of his letter on the sub- 
ject of Robert Burns, which expressed no stinted sympathy. Every one now 



1 88 Life of Sir William Rowan Hamilton. [1864. 



LETTER to PROFESSOR TAIT. 

T. 1858, p. 139, * ... It is not fair to ask my friends to be 
sincere about them [his sonnets]. That they have many faults I 
very well know ; and take a sort of pride (or perhaps vanity) in 
knowing it : because that point of self-knowledge appears to prove 
that I have kept too good company, personally and intellectually, to 
be satisfied with any poetical production of my own. But Francis 
Edge worth a Cambridge man, although an Irish one a younger 
brother of Miss Edgeworth, though by a different mother, who was 
during his life a great friend of mine, and for whose memory I 
retain respect and love, used indeed to criticise, very sincerely, some 
of my youthful verses ; but wound up by saying, " After all, 
Hamilton, your poetry will not disgrace you." 

I will not here refrain from stating my own opinion, strength- 
ened by that of friends specially competent to judge, that Hamil- 
ton's poems have, both in their diction and in their matter, qualities 
of enduring value ; that, speaking generally, they are, and will 
always be felt to be, fresh, graceful, fervid expressions of states 
of feeling and thought, interesting in themselves and possessing a 
heightened interest from their being the heart's utterances of a 
man of gigantic mathematical powers and of strong and deep 



admires Alfred Tennyson there is no merit now in praising him : but you will 
bear with me as an old friend while I say that I have lately been reading, over 
and over for I do not pretend to calculate how many times the " Princess." 
I may indulge the hope, at moments, that as I now read, with profit and 
delight, the book of the great Grecian Mathematician, Apollonius of lYrga, 
after an interval of two thousand years from its composition, so my own 
volume (of which I should be happy, if you thought you could manage it, to 
present Mr. Tennyson with a copy) may survive even several centuries nay, 
thnt, a.-, tin- earliest work in its own department, it may exist till books shall be 
no more. But it deeply presses on my reflection how much wiser a book is 
Tennyson's "Princess" than my "Quaternions" In saying all this 1 h <! 
I only echo what Wordsworth said to me while we were boating on 
Windermere in 1830 (I seem to see the splash of the oar). The words 1 il> not 
presume nor pretend to repeat; but the spirit certainly was, that in Alfred 
Tennyson, young a poet as he then was, there was a man of the highest 
promise. I am, my dear Robert, your very old and very all'ectiouate frit ml, 
William Rowan Hamilton.' 






AETAT. 58.] Mourey. 189 

affections ; and a few of them, it may be added, are so happy in 
thought and expression, as to claim their place in the poetry of 
his country, f As not irrelevant to this topic, it is also to be 
remembered that he found in Mathematics a field for the exercise 
of Imagination, and thus exemplified the connexion, which he was 
in the habit of asserting, between the highest Mathematics and 
Poetry. 

The same letter to Dr. Hart, from which I have just quoted, 
contains a reference to a recent correspondence of Hamilton's with 
Dr. Salmon, which may be given here as sufficiently summing up 
its purport to render unnecessary any further mention in this work 
of that correspondence. The passage has its value as a brief 
statement of steps in the history of double Algebra and Quater- 
nions. I regret that the lucid exposition by which it is followed, 
contrasting the methods of these diverse systems, is too abstruse to 
be here inserted. It is certainly worthy of reproduction in con- 
nexion with the very valuable letter on the same subject to Pro- 
fessor De Morgan, prompted by the same incident, and which was 
written on the preceding 25th of June : 

* ... In driving home yesterday I looked into the Second 
Edition (Paris, 1861) of Mourey's very ingenious little work. It 
was lucky that I could at once supply Salmon with a reference to 
the page of my Preface to the Lectures, in which I had cited the 
First Edition (Paris, 1828). But it is foolish to consider any such 
work as an anticipation of the quaternions. This brilliant and 
patriotic notion occurred lately to a French correspondent of our 
friend Salmon, who was so good as to send me the letter to read. 
The relation is rather of contrast than of resemblance, as in this 
very note to you I partly show. Systems which interpret + 1 dif- 
ferently cannot have much in common. I forget in what year it 
was that I first heard of Mourey ; nor is it of the slightest import- 
ance. As long ago as 1829 my attention was called by John T. 
Graves to the work of Mr. Warren, published in Cambridge, in 
1828, On the Square Roots of Negative Quantities. The systems of 
"Warren and Mourey, loth published in 1828, are substantially the 
same ; though the Frenchman was livelier and smarter. What was 



I go Life of Sir William Rowan Hamilton. [1864. 

"best in both had been anticipated in France by Argand, as I 
proved in the Preface above cited ; and the notion of an Algebra of 
the Plane as connected with "what we call Imaginaries" had certainly 
been thrown out by Wallis, in the 17th century, in England, as 
the same Preface shows. De Morgan has since elaborated a system 
of what he (I think happily) calls " Double Algebra," on the same 
basis as others before him, but with many new improvements of his 
own. 

6 Carrying then back this name, I conceive myself to have been 
acquainted with Double Algebra, through Warren, from the year 
1829. It cost me therefore fourteen years to unlearn that system, 
in order to pass from it to quaternions (in 1843). . . ,' 

On the 21st of August in this year Hamilton sent to Mr. Bab- 
bage a letter containing a full account of the system which he had 
invented some years before* for correspondence by cipher, and for 
which he claimed the merit of being able successfully to resist all 
attempts at deciphering without the use of the special key. Of 
this letter I have not been able to discover either draft or copy, 
but a large chart exists, made use of by the writer in preparing it. 
It seems expedient that this letter, if it can be recovered, should 
be placed, together with all connected documents, in the hands of 
some fit custodian. 

Towards the close of this year some commotion was excited in 
the scientific world by the circulation, with a request for signa- 
tures, of a Declaration, the object of which was to meet doubts 
thrown by the votaries of the Natural Sciences upon the "Truth 
and Authenticity of Holy Scripture " by an assertion on the part of 
Christian men of Science of their faith in the perfect reconcilable- 
ness of the results of scientific investigation with the statements of 
the sacred record. Sir John Herschel communicated to the Athc- 
ncBiim his protest against this movement as interfering injuriously 
with the freedom of thought of scientific men, and as drawing a 
line of invidious distinction, of which the social effects would be 

Supra, p. 42. 



AETAT. 5().] Declaration of Men of Science. 191 

pernicious ; and in the same periodical Professor De Morgan, with 
powerful logical analysis, exposed what he stigmatized as the un- 
fairness and sophistry of the Manifesto. Its social and logical 
faults having been thus dealt with by master hands, Hamilton, 
when subsequently applied to for his signature, sought to give his 
reply to the application, of which, equally with his friends, he dis- 
approved, as compendious a statement as was possible, and put 
forward its want of authority as his sufficient ground of refusal. 
His letter to the secretary was as follows : 

' OBSERVATORY, NEAR DUBLIN, 

1 December 23, 1864. 

* SIR, I received yesterday a circular, posted (as it appears) 
by you in London on the day before, in which I am requested to 
sign a certain declaration, known for some months to scientific 
men. It may be from my living so much out of London that I 
am entirely uninformed by what authority it is sought to impose 
this Fortieth Article of Religion : which, with all reverence for the 
Sacred Scriptures, I must decline to subscribe. I have the honour 
to be, sir, your obedient servant.' 

It may be admitted that the issuing of the Declaration was not 
ill-meant, but it will scarcely now be disputed that the measure 
was ill-judged. I refer the reader, in a note, to the parallel action, 
at a later time, of one well entitled to give, upon this subject, an 
influential opinion.* It is here sufficient to note that the part 



* The decision arrived at in regard to this application by Herschel and 
Hamilton, men who were sincere Christians as well as eminent in science, is 
vindicated in what appears to me to be a very wise view of the point in question 
expressed by one who takes rank with them both as a scientific man and a 
Christian James Clerk Maxwell on the occasion of his being asked, in 1875, to 
join the Victoria Institute. I quote from his Life, p. 404, the rough draft of 
the commencement of his reply : * I do not think it my duty to become a candi- 
date for admission into the Victoria Institute. Among the objects of the 
Society are some of which I think very highly. I think men of Science, as well 
as other men, need to learn from Christ, and I think Christians whose minds are 
scientific are bound to study Science, that their view of the glory of God may be 



192 Life of Sir William Rowan Hamilton. [1864, 

taken by Hamilton in the incident furnishes additional proof of 
the sturdiness with which he valued intellectual freedom, and was 
determined to assert it for himself and his fellow-workers in the 
fields of Science and Philosophy. 



as extensive as their being is capable of. But I think that the results which 
each man arrives at in his attempts to harmonize his Science with his Christia- 
nity ought not to be regarded as having any significance except to the man him- 
self, and to him only for a time, and should not receive the stamp of a Society. 
For it is of the nature of Science, especially of those branches of Science which are 

spreading into unknown regions, to be continually ' [Here the manuscript 

ends.] 



AETAT. 59.] The Last Year. 193 



CHAPTER L. 

PROFESSOR CAYLEY. HAMILTON'S LATEST SCIENTIFIC WORK. LAST 
ILLNESS. HONOUR FROM AMERICA. DEATH. 

(1865.) 

WRITING on the second day of this, the last year of his life, 
Hamilton begins thus a letter to his younger son: * It is a 
solemn thing, but I do not find it a painful one, to enter on a 
new year. I wish you many happy returns. It was my hope to 
have gone to Castleknock [to church] yesterday, but my cough 
was by no means so far gone as to make that safe.' These words 
may serve to indicate the religious ripeness of his spirit, and at 
the same time the shaken state of his bodily health, which from 
henceforth had to contend with a fatal combination of gout and 
bronchitis. To the same son, then serving as a curate at Clogher, 
he conveys his approbation of a sonnet, which on this account, and 
because of its upward-pointing significance, now specially con- 
genial to the father's thoughts and feelings, I place in a note.* 

* * TO A LARK SINGING OVER SNOW. 

Sweet bird, that nigh to Heaven's blue portal singest 

Above the snow-clad Earth, thyself unseen, 

And, free to roam where man not yet hath been, 

Sweet hope to us from unknown fountains bringest, 

A mild reproach to me thou downward fl ingest. 

What share hast Ihou in brighter days to come ? 

This prison Earth must be thy lasting home, 

Not that blue vault to which thou freely springest. 

While thankless I, with idle hands, sit dumb, 

Nor join the bird's glad song, the insect's hum ; 

While Heavenward my poor thoughts so seldom rise, 

So faintly knock, so hardly enter there, 

Thou hast the present entry of the skies, 

But / the Hope that makes those realms so fair. 

1 ARCHIBALD HEXRY HAMILTON/ 

VOL. III. 



194 Life of Sir William Rowan Hamilton. [1865. 

The following correspondence is upon an important mathema- 
tical theorem treated of by Hamilton in articles at the end of the 
Elements of Quaternions. I introduce it here mainly on account 
of its personal interest. It puts on record Dr. Salmon's judgment 
of what we may call the pcene-infallihility of Hamilton in mathe- 
matical calculation ; Hamilton's real unwillingness to be a cor- 
recting critic of the work of another mathematician ; his anxiety 
to do all justice to the powers and attainments of the great ma- 
thematician with whom he is thus brought into contact ; his active 
care to guard the rights in discovery of Mac Cullagh, and to ac- 
knowledge the anticipation by Mac Cullagh of a theorem he had 
thought to be his own; his expressed obligation to Professor 
Jellett for enabling him to do this ; his forwardness to make Mr. 
Cayley acquainted with all the facts ; and, finally, the knightly 
disposition of the latter to accept the rectification. 

The correspondence was begun by a letter from Dr. Salmon 
(Feb. 10, 1865), in which he writes : 

* ... I believe you know I am printing a second edition of 
my book on surfaces. I have just added a note referring to your 
explanation of the lines of curvature through an umbilic on a 
quadric. I perceive these are valuable things which you have 
added to the theory of twisted curves. . . .' 

Hamilton, in his reply (Feb. 13), after expressing his gratifi- 
cation, reports, in an important paragraph, his position in his own 
work (Elements of Quaternions] : 

' I am at more last specimens of physical applications of qua- 
ternions, and am treating briefly of Fresnel's Wave in the half- 
sheet, 5 B. But I have ready, in advance, as another specimen 
of physical optics, an article on Mac Cullagh's Polar Plane : after 
which I intend to devote scarcely more than a page to quaternion trans- \ 
formations of DJ + Df + D- 2 ,* and so to conclude my applications.' 

I have called this an important paragraph because, in con- 

ia D in this expression I suppose to be the symbol introduced at p. 710 
of the Elements of Quaternions. 



A i : i v T. 59. ] Concluding Articles of the ' Ekmcn A\ ' 195 

ncxion with what appears in his latest manuscript books, and in 
the final pages of the Elements as published, it proves that the 
last article in the book that on the Polar Plane was completed 
long before the article upon Fresnel's "Wave, which precedes it, 
and upon which he continued to be engaged in the month pre- 
ceding his death ; and because we learn from the clause which I 
have italicised, what was the one remaining, but unpublished, 
physical application of quaternions, with which he would have 
completed this series of illustrations of his calculus, before winding 
up, by general concluding remarks,* his laborious and gigantic 
treatise. Immediately after the passage I have just quoted, he 
proceeds : 

1 It was only last month that I was led to read Mr. Cayley's 
Paper in the Proceedings of our Academy, nor is the number be- 
side me at this moment ; but of course you can easily lay your 
hand on it, especially as you read the Paper for Mr. Cayley. 

* My analysis does not confirm his enunciation of the " Prin- 
ciple of Equivalent Moments ;" and I have been wishing to ask 
you two things : 

' I. Was that principle or property enunciated by Mac Cullagh 
hinuself, or was it deduced by Mr. Cayley from his principles ? 

' II. Has Mr. Cayley ever modified his enunciation of it since 
the reading of the Paper ? 

*I found that the reflected vibration should be considered as 
at the end of the reflected ray in the air; Mr. C. places it at the 
end of a prolongation of that ray in the crystal. This does not at 
all touch the mathematical merit of the Paper ; but it would make 
the greatest difference physically, in any consequences to be drawn 
respecting planes of polarization, &c. In short, I express the prin- 
ciple by an equation of three terms, and differ from Cayley by the 
xitjH of one of them/ 

Dr. Salmon, in his reply (Feb. 24, 1865), writes : 

* ... I have forgotten all about the polar plane. I was writ- 
ing to Cayley, and sent him on your criticism. I happened to 

* See Elements, pp. xiii., 495, 496. 
2 



196 Life of Sir William Rowan Hamilton. [1865. 

mention to a friend yesterday, as an illustration of the amount of 
my faith in you, " Sir William Hamilton writes me word that 
there is an error in a result of Cayley 's, and though he expresses 
himself with all possible modesty as to the possibility of there 
being an error or misconception on his side, still, without knowing 
anything more, I believe firmly that there is an error as he alleges. 
And if Sir "William were to tell me that a demonstration of my 
own was erroneous, I fear I should have no firm faith in any de- 
monstration of mine till his objection was withdrawn." 

On the 4th of March, Hamilton to Salmon continues thus : 

* I assure you that it is with great regret I find myself differ- 
ing in any point from Mr. Cayley, my admiration for whom has 
certainly sustained no diminution since I published, in 1847, my 
" First Series of Researches respecting Quaternions," in vol. xxi. 
part ii. of the Transactions of our Academy. In the last page of 
Note A, at the end of that Series or Essay, I expressed myself as 
follows (with respect to an early application by Mr. Cayley of 
Quaternions to the rotation of a solid body) : " That important 
application of the author's principles had indeed occurred to him- 
self previously ; but he was happy to see it handled by one so well 
versed as Mr. Cayley is in the theory of such rotation, and pos- 
sessing such entire command of the resources of algebra and of 
geometry." Nevertheless, having gone over my calculations again, 
and with great care, during the last few days, I cannot now feel 
any doubt of the existence of an oversight in the Paper which I 
mentioned last month. It was read by you (I remember that I 
could not attend that evening) on the 23rd of February, 1857, for 
Mr. Cayley, and is printed in vol. vi. part iv. of the Proceeding* 
of the Eoyal Irish Academy. I believe that I told you that I 
found the spherical trigonometry to be all right, when reading the 
Paper for the first time in January last, but that I objected to 
the sign of the moment of the reflected vibration, as given in pair" 
491 of the Part above cited. Recent calculations confirm this 
result ; but they go much further, for they show that when the 
reflected vibration is placed at the end of the reflected ray in air, 
the " Principle of Equivalent Moments " holds good for every < 
which is parallel to the face of the crystal? 

Not having gained from Dr. Salmon the historical information 



AETAT. 59.] Principle of Equivalent Moments. 197 

for which he sought, he turned, on the 6th of March, to Professor 
Jellett : 

' If you have received certain proof-sheets, which I indulged 
myself by sending to you recently, and if you will turn to the note 
at the foot of page 736, you will see that I have announced an 
intention of deducing, by quaternions, MacCullagh's "Theorem of 
the Polar Plane : " and in fact I have to-day made up for the 
printers, and am now sending to the post, my " Series 423," in 
which that deduction is given, with what seems to me great 
simplicity. But as I have also sketched a short addition to the 
Scries, which I am likely to send to-morrow, though it cannot well 
be in type this week, and which contains what appears to me a 
new theorem of physical optics the principles of Mac Cullagh and 
Neumann being used in the quaternion deduction I am anxious 
to consult you on the few following points : 

* I. Was the principle or property, called by Mr. Cayley the 
Principle of Equivalent Moments [Proceedings, E. I. A., vol. vi. 
p. 481, &c.], a deduction of Mr. Cayley' s own from Mac Cullagh' s 
principles, by him referred to ? 

* II. Or had the "principle of equivalent moments," as enunci- 
ated by Mr. Cayley, been actually deduced by Mac Cullagh himself 
(Mr. Cayley's modest tone suggests the latter alternative, but I do 
not know where to find any statement by Mac Cullagh on the 
subject) ? 

* III. Have you, since, either confirmed or corrected the enuncia- 
tion of the principle in question (given at p. 489 of the Paper) ? 

* IV. If not, do you know of anyone eke having done either ? 
For agreement with my " Theorem (or Principle?) of the Resultant 
Couple" rather recently arrived at the enunciation requires to 
be corrected by what amounts to a change of sign of the reflected 
vibration; or better, by considering the reflected rut/ in air, and not 
its prolongation in the crystal. 

' But even after thin correction which falls chiefly on the sign 
of the moment, given at top of page 491 the principle treated of 
by Mr. Cayley is only a very particular case of my new theorem, 
above referred to, of which I shall be happy to send you the 
statement, translated from the new quaternion formula." 



igS Life of Sir William Roican Hamilton. [1865. 

Professor Jellett's reply, dated March 9, was as follows : 

'The principle of the equivalence of moments is expressly 
stated by Mac Cullagh as a part of the general theorem by which 
he solved the problem of crystallized reflexion and refraction. 
You will find it in the twenty-first vol. of the Transactions, pp. 
41-2. 

' I confess that my interest in his whole theory has been much 
lessened by the discoveries of Jamin, which show that the theory 
can at best be accepted as only approximate, and in certain cases 
not even that. 

'I have not paid any attention to the subject at least to 
MacCullagh's mode of handling it for some time, and must 
therefore answer III. and IY. in the negative. 

' But I am not quite sure whether the verification you speak of 
be or be not experimental/ 

As his winding up of the correspondence Hamilton transmitted 
through Professor Jellett to Dr. Salmon, and through the latter 
to Professor Cayley, a manuscript-sheet containing a transcript of 
the last printed sub-article in the text, viz. (12), and the two notes 
appended on pages 761 and 762, of the Elements of Quaternions. 
In the second of these notes he refers to Mr. Cayley's " very clear 
and able Memoir," and after mentioning the " slight inadvertence 
in a Paper of such interest and value," repeats his " admiration 
(long since publicly expressed by him) which is due to the vast 
attainments of a mathematician so eminent as Professor Cayley. 3 
To this transcript was added, with the date March 13, 1865, the 
following note addressed to Dr. Salmon : 

' I hope that I shall be considered as not deficient in courtly 
to Mr. Cayley, to whom, if you choose, you can forward this shn-t. 
A letter received this morning from Jellett has given me a most 
useful reference to a later Paper by Mac Cullagh, m which I find 
myself supported, but of course anticipated, in what I had thought 
my own " Theorem of the Resultant Couple." So if I differ from 
Mr. Cayley I have Mao Cullagh on my side. My investigation 
was naturally quaternionic throughout ; and I have not caiu-t -ll< i <l 
any part of it, but merely made an Addition, as above, to the l:i>t 



AETAT. 59.] Professor Cay lc\. 199 

note of the series (423), which had already been sent to the printers, 
but will scarcely reach me in type before next week.' 

The sheet was transmitted by Dr. Salmon to Professor Cayley, 
by whom it was thus acknowledged in a letter addressed directly 
to Hamilton : 

' CAMBRIDGE, April 3, 1865. 

* I return with thanks the enclosed Paper, forwarded to me by 
Dr. Salmon. I can have no possible objection to the publication 
of it. I do not understand you to say that there is any error 
of sign in my theorem (2), but it is very probable that the true 
interpretation of this in reference to Mac Cullagh's theory should 
have been 

- Rt cos RU$mNUR + R"t" cos R" U" sin NU"R" 



= R't' costftf'sm NITR; 

cos 



- instead of + in the first term, and that in consequence I have 
wrongly enunciated the Principle of Equivalent Moments. I am 
not able to put myself back into the question to see that this is so, 
nor do I remember where my enunciation of the principle was 
taken from : I did not attend to the theorem otherwise than in a 
geometrical point of view, and was satisfied by obtaining the 
theorem (2) as a theorem in pure geometry, the interpretation of 
which should be the principle in question. I am therefore quite 
willing to admit that your correction is right.' 

The transcript, above mentioned, contains the whole of the 
concluding pages, from p. 760 (12) of the Elements of Quaternions, 
with the exception of the short sentence with which the last note 
concludes. This addition was founded upon the information 
which Professor Jellett's letter imparted. It is to be regretted 
that in preparing the manuscript for the press the latest form of 
this sentence was overlooked, as well as important following 
clauses, the main object of which was to obviate a possible in- 
ference from it to the disadvantage of Mac Cullagh. The passage, 
as finally settled, runs thus : * 

*See Manuscript Book D. 1864, p. 117. 



2OO Life of Sir William Rowan Hamilton. [1865. 

'The writer understands that subsequent experiments by 
Jamin and others are considered to diminish the physical value of 
the theory above discussed. But this neither detracts on the one 
side from the great intellectual and physical genius of Mac Cullagh, 
nor renders inappropriate on the other side, for the purposes of the 
present work, the recent pendant to the quaternion expression of 
Fresnel's earlier views.' * 

It is right I should here note that in the autumn of 1864 
Hamilton commenced a series of memoirs characterized by the 
use of determinants, which was carried on far into this the last 
year of his life.f 

On the 24th of March a date elsewherej assigned for another 
piece of work Hamilton received a letter from Dr. Charles Graves, 
at this time Senior Fellow of Trinity College, and Dean of the 
Chapel Royal, on the subject of Binomial Co-efBcieuts. Dean 
Graves wrote : 

' You know that the sums of the alternate coefficients in the 
binomial development are equal. . . . But is the corresponding 
theorem known respecting the three sums obtained by taking 
every third term in the series 

n (n - 1) n (n - 1) (n - 2) 



'' La' 1.2.3 ->' 

You will find that two of these are equal and the third differs 
from them by unity. The more general theorems are less elegant.' 

On the very day of the receipt of this inquiry Hamilton 
replied : 

' Your stated theorem was quite new to me ; I need not say 
that the suggested theorems were also unknown. But I have this 



See Appendix for additions to Contents of Elements of Quaternion*. 

t See Manuscript Book E., 1864, pp. 11 ... 40, 101 ... 116, 117, 119 

A prtci* of these memoirs is given in the Appendix. 

J See Appendix : * Memoirs in which Determinants arc used.' 



AETAT. 50.] Last Contributions to R. I. A. 201 

morning Lad the pleasure of proving the theorem which you enun- 
ciated by assigning general expressions for the three separate sums 
to which it refers. And I think that the analysis (by imaginaries 
and determinants) which I employed ought to extend to sets of 
four coefficients/ &c. 

On the next day he sends his proof of his friend's theorem, 
and on the 1st of April a generalisation, embracing all similar 
theorems. Dean Graves's theorem and proof, and Hamilton's 
generalisation, were subsequently communicated to the Royal 
Irish Academy, and may be found printed in its Proceedings for 
the meeting of June 26th, 1865. In the same number of the 
Proceedings is contained Hamilton's last addition to his long 
list of contributions to the scientific memoirs published by the 
Academy, of which he had been a member almost from his 
boyhood. It is entitled * On a New System of Two General 
Equations of Curvature ... all deduced from Gauss's 2nd method 
in his Disquisitiones generates circa Superficies CurvasJ and has 
appended to it ' Three Verifications of Measure of Curvature.' * 

Soon, however, Hamilton's wonderful activity of intellect, up 
to this time manifested uninterruptedly both in the carrying for- 
ward towards completion of the Elements of Quaternions, and in 
digressive exercitations suggested by sympathy in the work of 
friends, was brought to a pause by a severe attack of the illness 
from which no entire recovery was granted to him. 

* This Paper is additional to his treatment of the subject published in the 
Elements, 412, 413, and indeed is in continuation of a very extensive inves- 
tigation commenced on the 1 1th of January. 1864, and carried on at various dates 
up to April 25, 1864, as recorded in two folio manuscript-books (1864, No. 1, 
and 1864, No. 4), with the titles following: 'Lines of Curvature and Curvatures 
of Surfaces, partly by Quaternions, partly by the methods of Monge and Dupin,' 
38 pages, 130 articles. ' Gauss's Measure of Curvature of a Surface,' 2 pages, 
11 articles. 'Intersections of Normals to Quadrics,' 74 pages. 262 articles. 
4 Correspondence with Dr. Hart,' 13 pages. A third similar folio manuscript 
book (48 pages, 224 articles) is filled by an investigation, both by determinants 
and quaternions, of the * Locus of the Vertex of a Quadric Cone having Six- 
point contact with a Curve in Space,' with successive dates of work from Feb- 
ruary 1, 1864, to August 7, 1864. 



2O2 Life of Sir William Rowan Hamilton. [1865. 

On the 26th of April, 1865, we find him thus writing to a 
friend : 

' ... It is very kind of you to inquire about my health and 
spirits, and to tell me something of your own. Mine have not 
been good for (I may say) some years past ; but I am working 
away as usual. So much work produces naturally fatigue : in 
fact, it has injured my constitution, at least for the present ; and 
it is not particularly conducive to good spirits to find that I ha\v 
been running up what is (for my purse) a rather heavy bill with 
my printer.* But I have many blessings, and do not wish to be 
considered as a grumbler. . . .' 

On the 9th of May, Hamilton was in Dublin for the last time. 
On that day was opened the International Exhibition held in the 
Building now occupied by the Royal University of Ireland. It 
excited great interest in Hamilton : so much so that he declared 
his intention of coming in frequently to study its contents. But 

* In reference to this subject I grieve to say that it remained a weight 
upon his spirits up to the day of his death. On the 30th of May, 1865, the 
University Printer sent him an account, showing that the cost of paper and 
printing of the Elements of Quaternions, up to that date, was 395 12s. Id. 
Towards the discharge of this sum 200 had been paid 150 by the Bursar 
of Trinity College, and 50 from Hamilton's private purse thus leaving a 
balance against the still uncompleted book of 195 12s. Id. There remained, 
however, 50 still unexpended from College grants already sanctioned, so that 
145 was the sum still chargeable against Hamilton for the past; and to this 
an addition would have to be made on account of work subsequent to May 30. 
After his death, the Board of Trinity College liberally paid the above balance, 
together with the accrued addition just mentioned, and had then to defray 
the expense of binding, advertising, &c. The whole expense of printing and 
publishing amounted to close upon 500 ; and as the impression consisted of 
only 500 copies, many of which were presented to men of science, and scientific 
bodies although the book was soon out of print no profit could have resulted 
from its publication at the price of 1. I understand that so high a sum as 
5 has been given at a public sale for a single copy. For verification of these 
details I refer to Manuscript Book K, 1865, pp. 25-28. It contains the . 
of a letter from Hamilton to the Bursar of Trinity College, stating the a' 
fact*, and dated August 22, 1865 a date less than a fortnight distant from his 
death. One clause in this letter was, ' You will see that I have already paid 
him [the University Printer] Fifty Pounds of my own money a not \ 
pleasant operation.' 



A K i AT. 59.] Beginning of the End* 203 

this was not to be. An attack of acute gout in the lower limbs 
rendered it necessary for him to summon, on the 13th of May, the 
aid of Dr. Wyse, who continued to be in almost daily attendance 
for the remainder of the month, and who, as Hamilton's daughter 
records, was greatly struck by his patience under unusually severe 
guttering. It would seem that on the first three days of June he 
was apparently better, for the physician's visits were discontinued ; 
but on the 4th, alarming symptoms were manifested, and Dr. 
Stokes had to be sent for. The measures adopted were not suf- 
ficient to ward off an aggravated seizure on the following day, 
which took the form of epileptic convulsions. These were most 
severe, threatening immediate and mortal collapse. They were, 
however, under the direction of Dr. Stokes, at length subdued, 
leaving his strength finally shattered, and his mental powers for a 
time disabled. Before the end of the month he was again at work 
at his book, and corrected for the press his Papers for the Proceed- 
ings of the Academy ; but soon bronchitis supervened, and, with 
other ailments, led on to the inevitable close. 

It should not be here omitted that, during the continuance of 
this alarming and disabling illness, Hamilton and his family re- 
ceived from their neighbours, Mr. and Miss Eathborne, practical 
and effective sympathy of the most valuable kind. The two sisters- 
of Lady Hamilton had before this time passed away.* Scripples- 
town was now occupied by new inhabitants, but Dunsinea was 
still a family possession ; and Mr. John Garnett Eathborne suc- 
ceeding to his father, Mr. Henry Eathborne, proved always an 
attentive and attached neighbour to his aunt and her husband, 
His sister, Kate Eathborne, was the niece to whom, in the days of 
her opening beauty, Hamilton had addressed verses, printed in the 
second volume of this work.f Her intelligence and character de- 
veloped with her years, and she became to her uncle a favourite 
relative, regarded by him with constant interest and affection. 

It was just at this crisis that a letter reached him from across 
* See vol. ii., p. 2. t See vol. ii., pp. 61, 273. 



204 Life of Sir William Rowan Hamilton. [1865. 

the Atlantic, which shed brightness over his decline, and made 
him feel that, whatever might be the issue of his illness, his early 
dream of world- wide recognition was realised. It is remarkable 
that the land which first from a distance hailed with generous 
encouragement the beginning of his brilliant career was now to 
confer upon him a final crown of honour. When he was only 
twenty-seven, he received a diploma constituting him a Fellow 
of the American Academy of Arts and Sciences ;* now, when con- 
cluding his sixtieth year, and battling with mortal disease, he 
receives from the same great country the highest scientific honour 
in their power to bestow. It was after the tremendous conflict of 
North and South was ended, and both divisions of the continent 
had been by it welded into one nation, that the first truly national 
Academy of Science in America was formed. One of its earliest 
duties was to draw up a list of the most eminent scientific men 
throughout the world, upon whom the honour of Foreign Associate 
should be conferred. A discussion took place as to what name 
should be first on this list, and by a two-thirds majority it was 
decided that that place should be occupied by the name of William 
Rowan Hamilton. I give the letters of Mr. Gould, containing 
the details. I am sure that to the scientific chiefs of that noble 
nation it must be ever a deep gratification that the act of their 
Academy was in time to be welcomed by the dying mathematician, 
and they will learn with interest that the last letter written by him 
was in acknowledgment of it. It will be seen, indeed, that this 
letter, written only a week before his death, was a very inadequate 
acknowledgment of the distinction he had received ; but it was all 
his failing powers were capable of, and it intimated his intention 
of more fully expressing his feelings. A previous letter to Dr. 
JI;irt shows that those feelings were more than feelings of mere 
gratitude : that he was deeply impressed by the honour. Dr. 
Salmon's letter, which I subjoin, must have afforded him addi- 
tional gratification, as expressing personal sympathy, partitij 



See vol. i., p. 010. 



AETAT. 59.] Honour from U. S. of America. 205 

by many scientific and private friends, and the distinguished 
writer's sense of the honour reflected on the University of Dublin, 

From B. A. GOULD to SIR W. B.. HAMILTON. 

' CAMBRIDGE, MASSACHUSETTS, May 17, 1865. 

* DEAR SIR In the absence of Professor Agassiz from the 
country, the agreeable duty devolves upon me of announcing to 
you that the National Academy of Sciences, established by the 
United States on the 3rd of March, 1863, elected you, on the 9th 
of January following, first on the list of its Foreign Associates, 
now fifteen in number. 

' As no reply has been received to the notification directed by 
the Academy a year ago, it is feared that it may have failed to 
reach you ; and I therefore have the honour of addressing to you 
this duplicate announcement. A diploma will be transmitted 
hereafter. 

* The annual Eeports to Congress for 1863 and 1864, and the 
Academy's Annual for 1865, are this day sent you through the 
Smithsonian Institution. 

1 Allow me to add the expression of my personal gratification 
at the honour which the Academy has done itself in placing your 
name at the head of its foreign list, and of our hope for your 
cordial sympathy with our efforts to organize now, for the first 
time, a National Academy in the United States. 

' I am, my dear sir, very respectfully yours, 

' B. A. GTOULD, 

( Foreign Secretary pro tern. 9 

From the SAME to the SAME. 

' CAMBRIDGE, MASSACHUSETTS, May 20, 1865. 

' MY DEAR SIR In August or September next it will have- 
been twenty years since within ten days after first setting foot 
on European soil I found my way to you, and was received with 
a cordial welcome and kind hospitality, which might well have 
gratified a man of established scientific fame, instead of a youngster 
under twenty, who had never seen much of the world on this side, 
or any of it on your side the Atlantic. 



2o6 Life of Sir William Rowan Hamilton. [1865. 

* I cannot tell how often my memory lias reverted since then 
to your heautiful park and your pleasant house and cordial 
greeting, and hoped that it might yet be my good fortune to pay 
my respects to you once more. The lapse of time seems, however, 
to make it more difficult with each successive year to leave the 
various duties which seem to bind us closer and closer to our 
homes and domestic routine of duties, and it is doubtful whether 
I may ever have the hoped-for gratification. 

* But I cannot resist the impulse to send these few lines while 
waiting the accompanying official document, and to express my 
hope for the long continuance of your health and scientific activity. 

1 Our new Academy is the first national or governmental insti- 
tution we have ever had in the United States. All other scientific 
societies have been purely voluntary associations. This one was 
founded in the midst of the severest trials to which a nation was 
ever subjected without destruction; and it is not the least of our 
sources of pride and hopefulness in its behalf, that it dates from 
such an epoch and crisis in our country's existence. You can 
easily imagine that the difficulties to be overcome in the United 
States are very different, as well as very much more serious, than 
those which would be encountered in an older country or in one 
whose form of government gave the chief power to the most culti- 
vated and highly educated class. It is to the representatives of 
the people that we must look for support and aid ; and these are, 
in general, just what their name indicates representatives of the 
people, neither better nor worse than the average. But as they 
all mean to do in these matters what is best for the whole country, 
it is only needful to convince them of what is requisite, though this 
is not always an easy matter. 

' In January, 1864, the Academy voted to elect ten Foreign 
Associates, and accordingly twenty names were agreed on ; aft t r 
which the discussion turned, not on the election or non-election of 
any one of them, but upon the order in which they should be 
inscribed upon the rolls. It was soon narrowed down into a dis- 
cussion as to whether your name or that of Professor von Baer, of 
8t. Petersburgh, should head our list ; and the Academy finally 
decided the question by a vote of two-thirds against one. Whetln-r 
the notifications then sent by Professor Agassiz were lost or mis- 
laid, or Buffered shipwreck, we cannot learn. But they have none 






AETAT. 59.] Honour from National Academy of U. S. 207 

of them apparently reached their destination : certainly no re- 
sponses have been received, and Professors Argelander, Milne- 
Edwards, and Brown have certainly not received those sent to 
them. Therefore they are all sent anew by the present mail. 

* Please accept my sincere respects and best wishes for your 
continued health and usefulness, and believe me, my dear sir, ever 
most faithfully yours.' 

From SIR W. E,. HAMILTON to A. S. HART, LL.D., S.F.T.C.D. 

1 OBSERVATORY, June 13, 1865. 

' MY DEAR DR. HART It was not until to-day that I read an 
official letter from America, which reached me more than a week 
ago, and which I now enclose for your perusal, and, if you think 
that it could interest them, for that of the Board also. To have 
been elected last year out of the whole world by the new National 
Academy of America, and, as a private letter informs me, by a 
majority of two to one, the^rs^ of its Foreign Associates appears 
to me so surprising a thing, that I might be apt to treat it as 
incredible, if I had not been long acquainted with the writer of 
the communication. 

* To the indulgence of my College patrons the result may 
appear less extravagant than to myself. I shall wish to have the 
letter returned at your convenience. 

' 1 have still papers connected with my book to enclose to you, 
but for the moment can only subscribe myself, dear Dr. Hart, very 
truly yours. 

4 P.S. I am considered to be slowly recovering, and am able 
to work a little.' 

From GEORGE SALMON, D.D. to SIR W. E. HAMILTON. 

' T. C. D., June 16, 1865. 

' MY DEAR SIR WILLIAM I have heard with very great plea- 
sure of the honour done you, and through you to our University, 
by the National Academy of America. I am the less surprised, 
however, as I had heard some years ago from Professor Peirce of 
the extent to which your Lectures on Quaternions circulated and 
were appreciated in America. 



208 Life of Sir William Rowan Hamilton. [1865. 

' If this day week (Thursday) should be fine, I should like to 
go out to see you. As, however, I only intended a friendly visit 
of inquiry, do not have the least scruple in telling me if, when the 
time comes nearer, you do not feel yourself quite equal to receiving 
visitors. If you were living in town we should, no doubt, all be 
calling every day to ask how you were going on. But, as you are 
not so accessible, it is natural to prefer to make my inquiries by 
letter, unless there is a hope of seeing you. I remain very sin- 
cerely yours.' 

From SIR W. E. HAMILTON to B. A. GOULD. 
[FROM A COPY.] 

* OBSERVATORY, August 24, 1865. 

* MY DEAR MR. GOULD I have been prevented by illness from 
writing sooner to acknowledge the high compliment paid to me 
lately in America, and announced in your letter. I am anxious 
not to let another post pass without doing so, and shall write 
afterwards more fully. I am, &c.' 

On the 16th of July I went out to the Observatory to visit my 
old friend. I had not seen him since the 4th of May. I had 
heard of his being present at the opening of the International 
Exhibition, but the tidings of his severe illness was slow in reaching 
me. A false rumour, indeed, of his having been again in town led 
me to invite him to meet my brother-in-law, Leopold von Ranke 
(then in Dublin to receive from our University an honorary 
degree). The contents of his reply, written * before daybreak' 
on July 4, and the infirm handwriting, undeceived me as to his 
condition ; but illness of my own and incessant occupation pre- 
vented me from making earlier personal inquiry. I was greatly 
shocked by the change which had taken place in the interval. 
Emaciated and feeble, he seemed altogether in physical respects 
a different man. His intellect, I soon found, was as clear and 
active, and powerful as ever, and I was deeply impressed by th*> 
gracious sweetness, gentleness, and humility, which shone through 
his manner and every word he uttered. He spoke with thankful- 



AETAT. GO.] Last Illness. 209 

ness of his having been brought through the struggle for life which 
he had undergone, and of the expectation thus revived that the 
great task he had on hand would very soon be accomplished by 
his sending to the press the last sheets of his Elements of Quater- 
nion*. And looking beyond this event, he intimated his intention 
of turning afterwards, for refreshment and variety, to the study of 
poetry and to the putting down of his thoughts on metaphysics. I 
spent with him nearly four hours in conversation, embracing the 
higher topics of the day, religion included, part of the time in his 
beloved garden, part in the house, from which with kind considera- 
tion he insisted on sending me homeward in his car ; and I felt, as 
I drove away that summer evening from the Observatory, that 
never in the long period of our friendship had my feeling towards 
him been one of deeper admiration and affection. 

On the 5th of August Hamilton commenced what was intended 
to be a short series of letters on Quaternions, addressed to his 
younger son, by writing with a tremulous hand the letter describ- 
ing the circumstances of his great discovery which has been printed 
in the second volume of this work.* An introductory sentence to 
a second letter of the series is all that he was able afterwards to 
accomplish. In the Appendix of this volume will be found some 
paragraphs carrying on the Contents of his book, which were 
written towards the close of the month. The 26th of August is 
the date attached to the last of these paragraphs, and this, I 
believe, is his latest registering of scientific work. On the 22nd 
he wrote to the Bursar of Trinity College his letter already re- 
ferred to,f respecting the University printer's account, and on the 
24th of August to Mr. Gould his brief acknowledgment of the 
honour wafted to him over the Atlantic. These were his final 
efforts. 

On the 2nd of September, in response to a summons from his 
eldest son, I proceeded from Howth, where I was then staying, 
to the Observatory. I was met at the door by Lady Hamilton in 

* Vol. ii., p. 434. f Supra, p. 202. 

VOL. III. P 



2IO Life of Sir William Rowan Hamilton. [1865. 

tears, and from her and Mrs. Joseph Willey, who was rendering 
valuable aid as nurse to her relative, and from his eldest son* I 
heard the sad tidings that the end was at hand. I found my 
friend in bed, his breathing rendered difficult by bronchitis, but 
his mind calm and in its full strength. He at once disclosed his 
consciousness that he was approaching the termination of his life. 
After referring to his long friendship with myself and other mem- 
bers of my family he spoke on general topics of a religious bearing, 
taking occasion, as I have already intimated,! to express his dis- 
approval of Bishop Colenso's treatment of Abraham's interrupted 
Sacrifice of Isaac, and expressing his conviction that God, as 
supreme Lord of life, able to give, to take away, and to restore, 
might quite justifiably give the command impugned : though he 
considered not unlikely, what I urged, that the whole transaction, 
besides being a test of Abraham's faith, might be rightly inter- 
preted as a process of symbolical instruction, one lesson of which, 
though not the highest, was that human sacrifice was not accept- 
able to God as part of human worship. Turning to his own 
relation to God, he asked me a question by which I was deeply 
moved, manifesting, as it seemed to me, his humble searching of 
heart, * Did I think that God could love him ? ' I replied as the 
Christian minister has the happiness of being able to reply to such 
inquiry from a God-loving, yet self-arraigning, self-condemning, 
spirit, and he was satisfied. He then asked me to pray with and j 
for him, telling me that he had found in the 145th Psalm, which 
he had asked Mrs. Willey to read for him, a wonderfully suitable 
expression of his thoughts and feelings and truly in that 
we may read his admiring thoughts of God and God's works, 
feelings of gratitude for the mercy that had sustained him through | 
life, upheld him in falls, and listened to his cry in sorrow. I com- 
plied with his desire ; after which he said that he wished to testify 



His younger son and his daughter were now at Clogher : the former serv- 
ing his cure, the latter recruiting her strength after illness. 
t Vol. i., p. 179. 



A i :T AT. 60.] Deal '/I. 211 

! his faith and thankfulness as a Christian by partaking of the 
Lord's Supper. He added words to this effect that, personally, 
he would rather receive it at my hands, or those of his son, than 
from anyone else in the world, but that he thought the rubrical 
direction of our Church should be observed (thus manifesting 
at the last his deeply-seated respect for order and law), and that 
he would therefore ask me to request his Parish Clergyman, 
Dr. Sadleir, to come to the Observatory on the next day, if 
possible, for the purpose of administering the sacred rite, and 
that I would join him in partaking of it. I, of course, consented, 
and withdrew, as the arrival of Dr. Stokes and Dr. Wyse was 
announced. He then roused himself and used for the last time 
his pen so long his almost inseparable instrument of thought 
in feebly writing a few words to prepare his physicians for finding 
him with little voice left after his long converse with me. Upon 
their coming down from his room I gathered from them that he was 
indeed come to the final stage of his illness, but that it could not 
be pronounced how long his powers would hold out. Dr. Stokes, 
who was at the time my medical adviser, would not suffer me to 
remain, as I wished to do, but insisted on taking me back with 
him to Dublin. I therefore went again to my friend's bedside, 
and was struck by the signs of a great collapse of vital energy, 
but had the satisfaction of a momentary interchange of farewells. 
Very shortly after our departure, at 2h. 30m. p.m., he breathed 
his last, having first, as I learned the following day, solemnly 
stretched himself at his full length upon his bed, and symmetri- 
cally disposed his arms and hands, thus calmly to await his 
death. 

News of the event reached me the next morning, accompanied 
with a request, immediately acted upon, that I should revisit the 
mourning family. I then beheld, for the last time, the form of 
my friend, and I shall never lose the impression of grandeur and 
majesty made upon me by the noble head and the monumental 
figure. More than once, in after times, Dr. Stokes has said to me 
that in all his long experience he had never seen a human being 

P 2 



212 Life of Sir William Rowan Hamilton. [1865. 

whose aspect, under the suhliming power of death, could compare, 
in the attributes I have named, with that of Hamilton. 

The Provost of Trinity College, Dr. Eichard MacDonnell, 
writing on the 4th of September, expressed, in the following words 
of a letter to Hamilton's son, his estimate of the event viewed in 
.connexion with the history of the University of Dublin : * Accept 
my heartfelt condolence for the great loss which your family has 
sustained. The loss to our University is irreparable, for since the 
oreation of our University we have not had so distinguished a 
member of our body.' 

The funeral took place on the 7th of September. His remains 
were brought from the Observatory to the Chapel of Trinity Col 
lege, where the first part of the Burial Service was read by his 
friend, Doctor Todd, Senior Fellow ; thence a procession followec 
the hearse to the cemetery at Mount Jerome. The time of yea 
accounted for the absence from Dublin of very many who woul( 
have joined it. I may name two who were, with great regret, a 
a disabling distance his early friend and pupil, the Earl of Dun 
raven, and Dean Graves. Still great numbers formed a proces 
sion, which included, besides his family and connexions, a large 
body of College students, headed by the University authorities 
many citizens of Dublin, preceded by the chief officers of the 
Corporation of his native city ; and members of the Royal Irish 
Academy, attended by their secretary, Mr. Clibborn, bearing the 
Academic mace veiled in mourning reminiscence of their grea 
president. I might here give a list, not uninteresting, of eminen 
men who paid . this tribute of their respect, but must conten 
myself with mentioning that among them were Hamilton's old 
friends, Dr. Hart, Dr. Salmon, Dr Stokes, Dr. Petrie, Sir Thomas 
Laroom, Denis Florence MacCarthy, and Professor Adams o 
Cambridge, The conclusion of the Burial Service was read lr\ 
Dr. Sadleir, Incumbent of Castleknock. The grave appropriate] 
to Hamilton, on the north side of the . demetey, is marked by o 
headfltone, subsequently erected by his family, and bearing the 
insertion : 



AETAT. 60.] B Until. 213 



HERE LIE THE MORTAL REMAINS OF 
SIR WILLIAM ROWAN HAMILTON, LL.D., 

110 YAL ASTRONOMER OP IRELAND. 

HE WAS BOEN AUG. 4 : 1805. 
HE DIED SEP. 2 : 1865 

' IN THE LOVE OF GoD, LOOKING FOR THE MERCY OF OUR LORD JESUS CHRIST 

I'NTO ETERNAL LIFE.' Jlldt 21. 



It should here be recorded that the Civil List Pension of 200, 
which had been conferred on Sir W. E. Hamilton in 1843, was 
continued, after his death, to his widow and daughter conjointly. 
Lady Hamilton died in 1869, his daughter, Mrs. O'Regan, in 
1870, when the pension lapsed. An effort was then made to ob- 
tain a grant from the same source for his only surviving sister, 
Sydney Margaret Hamilton, a lady who would in every respect 
have been a fitting recipient of the favour for she had in many 
ways aided her brother, both in domestic emergencies, and in the 
routine work of the Observatory, was of exemplary character, and 
was at this time in very narrow circumstances. But though a 
memorial, urging the grant, was signed by the chief men of 
science and literature in Ireland, the Premier of the time, Mr. 
Gladstone, found himself unable to accede to its prayer. Miss 
Hamilton afterwards emigrated to New Zealand, where she still 
lives, esteemed and honoured. The elder son of Sir William, Mr. 
W. E. Hamilton, has become a citizen of the Dominion of Canada, 
and the younger son is a laborious and respected, but unbeneficed, 
clergyman in the North of Ireland. Sir William's only grand- 
child, John Rowan Hamilton 0' Regan, born but a few weeks 
before the death of his mother, is now a promising scholar of 
Clifton College. 



214 Life of Sir William Rowan Hamilton. [1865. 



CHAPTER LI. 

s 

CHARACTERISATIONS OF HAMILTON AND OF HIS WORK. 

THE family of Sir "William Hamilton received in their affliction 
many letters of condolence. Two of these, addressed to his widow, 
are here inserted, inasmuch as the writers, Sir John Herschel and 
Augustus De Morgan, were eminently qualified to form an ade- 
quate judgment of the intellect, the scientific work, and the 
character of the friend whom they had lost, and it would be 
unfair, I think, both to Hamilton and to the public not to place 
that judgment on record. 

From SIR J. F. W. HERSCHEL to LADY HAMILTON. 

' COLLINGWOOD, September 14, 1865. 

* DEAR MADAM I trust that, though personally a stranger, I 
shall not be thought obstrusive in offering my most sincere and 
heartfelt condolence on the occasion of your late heavy affliction, 
as one who loved and admired him whom it has pleased Providence 
to remove (while yet at an age when his country and science 
might still have expected so much from him) with no common love 
and admiration. 

* The event, which I learned only from its public mention in 
the papers, struck me with the more painful surprise, as I had not 
heard of his previous illness, and have been lately several times on 
the point of writing to inquire respecting the progress of that really 
important work on which I knew that he was engaged an \- 
position, in a form adapted to the comprehension of ordinary 
mathematicians, of the principles of his wonderful Cdlcu/it* <//' 
Quaternions. I trust that he left this work, if not quite com- 
pleted for press, at least so far finished that it may yet see the 
light. I will not intrude longer on your sorrow. Permit me only 
to add that among the many scientific friends whom time lias 



1865.] Tributes from Herschel and De Morgan. 2 \ 5 

deprived me of, there has been none whom I more deeply lament, 
not only for his splendid talents, but for the excellence of his dis- 
position and the perfect simplicity of his manners so great and 
yet so devoid of pretension and allow me to remain, dear madam, 
your faithful servant, J. F. W. HERSCHEL. 

' P. S. I need not add that Lady Herschel desires me to add 
the expression of her hearty condolence.' 

From PROFESSOR DE MORGAN to LADY HAMILTON. 

1 91, ADELAIDE-ROAD, N.W., September 5, 1865. 

1 DEAR LADY HAMILTON I shall certainly not intrude on your 
grief by sending this letter according to its date ; I write it on the 
day on which I have received the shock of hearing that one of my 
dearest friends has passed away, without my knowing that any the 
least alteration had taken place in his health. That I shall sooner 
or later offer you my heartfelt sympathy is a thing of course ; and 
I cannot do it better than by writing at once, though I know the 
sending it at once would be wrong. 

* Several months have passed since I heard from your husband. 
This was not uncommon ; our correspondence was often interrupted 
by longer periods ; it went and came by fits and starts. I hoped 
that all things were going on as usual, and was meditating a letter 
of inquiry to be replied to as many of the same kind had been 
before, when three lines in the Daily Telegraph put me in posses- 
sion of the sad news that nothing of my friend was left in this 
world except his deathless fame and his mourning family. 

' I have called him one of my dearest friends, and most truly ; 
for I know not how much longer than twenty-five years we have 
been in intimate correspondence, of most friendly agreement or 
disagreement, of most cordial interest in each other. And yet we 
did not know each others' faces. I met him, about 1830, at 
Babbage's breakfast-table, and there, for the only time in our 
lives, we conversed. I saw him, a long way off, at the dinner 
given to Herschel (about 1838) on his return from the Cape; 
and there we were not near enough, nor, on that crowded 
day, could we get near enough, to exchange a word. And 
this is all I ever saw, and, so it has pleased Grod, all I shall 
see in this world, of a man whose friendly communications were 



2l6 Life of Sir William Rowan Haw ilton. [1865. 

among my greatest social enjoyments and greatest intellectual 
treats. 

* There is not a word which I could offer to you or yours, on 
any of those considerations which bring such comfort as the case 
allows, other than what must suggest itself, and must be better 
enforced by those around you. That you should soon find comfort 
in such considerations would be nay, h his chief wish for you 
and your children. May you find it ! 

' The time will come when you, or Miss Helen, will feel able to 
write me something about his last months, his declin.3, and die state 
in which his scientific matters are left. You will, I sincerely hope, 
not hurry in compliance with my request. No number of months will 
abate my interest in the matter. I have watched his career from the 
beginning. When I a year younger than himself, as it happens 
was an undergraduate not far advanced, and he must have been 
about nineteen years old, I heard of the extraordinary attainments 
of a very young student of Trinity College, which were noised 
about at Cambridge. This, rumour was made more interesting by 
other rumours which also circulated about the same time concern- 
ing another young Irishman, then recently matriculated at Cam- 
bridge. This was poor Murphy, whose subsequent career, though 
great in mathematics, fell short in conduct and discretion. He 
wanted all but mathematical education in early youth. The 
appearance of the two at once in the field gave both an interest, 
and I was thus led to watch Hamilton's career before I knew 
anything of him personally. 

* His memory will be very bright and very lasting. I trust 
that care will be taken to illustrate the singular variety of his 
attainments and the fertility of his mind. His publications give 
no more than a glimpse of what he was out of mathematics. Nor 
must it be left unrecorded how truly good he was as a man and a 
member of Society. In exact science he will be the Irishman of 
his day and of that to come, just as much as his namesake was in 
mental speculation the Scotchman. 

* With th< warmest wishes for you and yours, I am, dear Lady 
Hamilton, sincerely yours, A. DE MORGAN.' 

In ;i brief memoir of his friend, printed in the Gentleman * 
r .January, 1806, Professor De Morgan bears additional 



1865.] De Hfortftifs Obituary Nutia. 2 I 7 

totimony to the moral and intellectual character of Hamilton, and 
furnishes some illustrative anecdotes and details which have a 
v:ilue, as helping to produce a life-like image of the man. The 
following are extracts : 

1 .Hamilton was a man who combined different talents to an 
ut which is often attributed, by exaggeration, to the possessor 
of one powerful faculty ; bnt in his case there is abundant evidence. 
He was scholar, poet, metaphysician, mathematician, and natural 
philosopher. Highly imaginative and fluent of tongue, he was an 
orator in all that he knew ; even in mathematics, to the details of 
which he could give almost a rhetorical cast in a letter. In meta- 
physics he was very well read, and could talk in a way which 
suggested to Southey a comparison and a difference. Hamilton 
one day preached to Southey on this subject, until the latter 
remarked, as they passed a ploughman, " If you had been Cole- 
ridge, you would have talked to that ploughman just as you 
have been talking to me. . . ." 

* Hamilton was not only an Irishman, but Irish ; and this with 
curious oppositions of character. He was a non-combatant : there 
was too much kindness in his disposition to allow any fight to show 
itself. Impulsive and enthusiastic, with strong opinions and new 
views, he was never engaged in a scientific controversy. . . . 
William Eowan Hamilton's preservative was his dread of wound- 
ing the feelings of others. In his youth " Defender of the Absent " 
was his nickname . . . He had a morbid fear of being a plagiarist ; 
and the letters which he wrote to those who had treated like subjects 
with himself sometimes contained curious and far-fetched misgivings 
about his own priority. But with all this there was a touch of the 
national temperament in him ... an Irishman who never gets into 
a row may give quick but quiet symptoms of opposition of opinion, 
and of what, were it more than a rudimeut, would be called pug- 
nacity. 

* Hamilton was apt to work by fits and starts. He has been 
known several times to work fourteen hours in one day, standing 
nearly all the while; but there were intervals of comparative 
inaction . . . Sometimes a letter was written and copied which was 
not sent for months, and then only the first sheet, with promise of 
the rest. It has even happened that the letter was knowingly 



2l8 Life of Sir William Rowan Hamilton. [1865. 

never forwarded at all, and that when, long after, he found reason 
to wish to send it, he could not find it and sent the copy instead. 
But with all this he made more notes than anyone, and was 
exceedingly particular about minute accuracy of points, crosses, 
and dates in the most trifling memoranda. His first Lecture* on 
Quaternions, to our knowledge, had a dozen sheets printed off by 
December, 1851, and appeared only in the middle of 1853 ; the 
second set, which will probably appear by the time this account is 
printed, took a much longer time in passing through the press. 
The proof-sheets were held in hand until the author had satisfied 
himself about himself, and about others : and neither was easy 
work. . . . His papers were in most picturesque confusion, but he 
knew how to lay his hand on any one he wanted [not always] ; he 
could detect the removal, were it only by a quarter of an inch, of 
any out of hundreds, and any such offence against the laws of his 
study would throw him into what our informant calls a " good, 
honest, thundering passion.". . . 

* In the matter of right and wrong, Hamilton was very simple- 
minded. To say he was truthful would be only a part of the truth ; 
his aptitude to entertain misgivings, already alluded to, made him 
often think it right to express his opinions to avoid the possibility 
of being misunderstood. But it may be said that it was not he 
and others who differed, but his opinions and the opinions of 
others; his tolerance was perfect. . . . 

* He very much liked Goldsmith's writings, and we thiuk 
points of similarity might be traced between him and the author 
whom he so much admired. But the parallel would break down 
altogether in one point ; Hamilton spoke as well as he wrote. 
His voice was distinct, sweet, and powerful. He relished the 
extremes both of simplicity and splendour, though in his own 
habits and manners as plain as possible. He thought much of 
the comfort of others and lightly of his own. When some house- 
breakers were caught on the premises, and detained until they 
could be carried before a magistrate, he amused his family by 
directing that the felons should be asked whether they prefenv.i 
tea or milk for breakfast. A full memoir of his private aud 
public life would present a genial combination of intellectual 
greatness, moral goodness, and piquant peculiarity of thought ami 
manner, all brightened by never-ceasing benevolence of fooling, 
and toned by rare gentleness of manner.' 



1865.] Hamilton the Irish Lagra* 219 

In reference to the combination in Hamilton of poet and 
mathematician, De Morgan adds: 

* The moving power of mathematical invention is not reasoning, 
but imagination. We no longer apply the homely term maker in. 
literal translation of poet ; but discoverers of all kinds, whatever 
may be their lines, are maker* ; or, as we now say, have the 
creative genius. 

* Hamilton was once called the Irish Lagrange, and the com- 
parison was a good one. The styles of mathematicians differ as 
much as the styles of poets; and Hamilton is distinguished by 
that power over symbols, combined with elegance of expression 
which is so remarkable in the writings of Lagrange.' 

From the Eloge Ilistorique, pronounced by Fourier on Laplace 
in the Academic Roy ale ties Sciences, on the 15th of June, 1829, I 
extract the passage in which the great mathematician who delivered 
it describes the work and character of Lagrange. It will be found,, 
I think, largely to justify the appellation above recorded as applied 
to Hamilton. 

* Le trait distinctif de son genie consiste dans P unite et la gran- 
deur des vues. II s'attachait en tout a une pensee simple, juste, et 
tres-elevee. Son principale ouvrage, la Mecanique Analytique, 
pourrait etre nominee la Mecanique Philosophique, car il ramene- 
toutes les lois de I'equilibre et du mouvement a un seul principe ; 
et ce qui n'est pas moms admirable, il les soumet a une seule 
methode le calcul dont il est luimeme 1'inventeur. Toutes ses 
compositions mathernatiques sont remarquables par une elegance 
singuliere, par la symetrie des formes et la generalite des methodes, 
et, si Ton peut parler ainsi, par la perfection du style analytique. 
Lagrange n'etait pas moins philosophe que grand geometre. II 
1'a prouve, dans tout le cours de sa vie, par la moderation de ses 
desirs, son attachement immuable aux interets generaux de 
riiumanite, par la noble simplicite de ses moeurs, et 1'elevation 
du caractere, enfin par la justesse et la profondeur de ses travaux 
scientifiques.' 

At the Stated General Meeting of the "Royal Irish Academy r 



22O Life of Sir William Rowan Hamilton. [1865. 

held on the 30th of November, 1865, Hamilton's Eloge was pro- 
nounced by the President of the Academy, Dean Graves, now 
Bishop of Limerick. The Eloge is printed at full in the Proceed- 
ings of the Session, but I think it right to reproduce here passages 
from it which furnish interesting particulars not elsewhere dwelt 
upon as to the modus operandi of Hamilton in carrying on his 
scientific work, and convey on other important points testimony 
valuable and implicitly to be trusted, as coming from one who, 
having been long his scientific associate and private friend, was a 
specially competent witness : 

* Hamilton was gifted with a rare combination of those qualities 
\vliich are essential instruments of discovery. He had that fine 
perception of analogy by which the investigator is guided in his 
passage from the known to the unknown. This is an instrument 
by which many important mathematical discoveries have been 
effected. Sometimes the mathematician devises some happy modi- 
fication in the statement of a theorem or a method, by which its 
application may be extended. Sometimes, by analyzing different 
demonstrations, he even sees that a particular proposition may be 
made the starting-point from which he ascends to more than one 
generalization. In the investigations of Hamilton we find abund- 
ant instances of the skilful use of all the ordinary expedients and 
instruments of inventive sagacity. But he seems, also, to have 
possessed a higher power of divination an intuitive perception that 
new truths lay in a particular direction, and that patient and 
systematic search, carried on within definite limits, must certainly 
be rewarded by the discovery of a path leading into regions hitherto 
unexplored. Something like this was the unshaken assurance which 
led Columbus to turn his back upon Europe, to launch upon the 
broad Atlantic, and seek a new world in the far-off west. 

'And our illustrious countryman's diligence in research was 
not less admirable than his prescient sagacity. No amount of 
labour to be incurred could deter him from entering upon the 
calculations by which the correctness of his conjectures was to be 
tested. The confident expectation of obtaining results instructive 
in one way or undhcr ivrnneilrd him to the irksomene.^s ..!' the 
most tedious and COinplicat*-.! mU-ulations. lie felt that lln' gi 






1865.] Dean Graves s Eloge. 221 

object to be sought, in the first instance, was the discovery of the 
result itself ; and he trusted that, once it was reached, he would be 
able to strike out some more direct and more elegant method of 
investigation. His MSS., even his published researches, furnish 
many examples of this. Once he had reached the conclusion at 
which he had been aiming, he resumed the consideration of the 
principal steps in his argument ; he interpreted them with care ; he 
traced their connexion, and seldom failed to arrive at simplifications 
and generalizations, which amply compensated for the labour spent 
upon his first essays. By this habit of grappling courageously 
with the difficulties of calculation he was distinguished from some 
other eminent mathematicians. Averse to plunge into depths of 
calculation from which they see no certain hope of emerging in 
the end, they are tempted to expend an undue amount of intel- 
lectual energy in the endeavour to force their way by a direct 
method to the desired result. . . . 

c In the case of Hamilton, it is, moreover, deserving of notice 
that he evinced a readiness to grapple with the difficulties of 
calculation, even where there was no prospect of his labour being 
rewarded by any discovery. He engaged in exercises of this kind 
sometimes from a wish to strengthen his intellectual hold of gene- 
ral propositions by scrutinizing the results obtained by applying 
them in a number of particular instances ; and sometimes, perhaps, 
from a wish to mature and keep in exercise those powers of calcu- 
lation upon the exactitude and prompt operation of which so much 
depends in the conduct of difficult mathematical investigations. I 
have known him spend hours, or even days, in working numerical 
examples of some theorem in pure or applied mathematics, or in 
testing the accuracy of some formula of approximation. Occasion- 
ally he engaged in tasks of this nature, in the kindly endeavour to 
convince some half-crazed squarer of the circle that his proposed 
construction was inaccurate. Finding almost always that it was 
hopeless to convince the mathematical fanatic of the unsoundness 
of any of his premises, he would take pains to show him that the 
results he obtained were false in particular instances. 

4 And this leads me to notice a feature in his character which 
deserves to be recorded. From the lofty height of his genius and 
learning he was accustomed to stoop with the utmost readiness to 
hold converse with inferior minds. Many of his visitors at the 



222 Life of Sir I J lllia m Rowan Ham ilton . [ 1 865. 

Observatory, and the members of the class who attended his 
lectures in Trinity College, can recall instances of his patience 
and good-nature in answering their questions, and clearing up 
the difficulties which beset them in their elementary studies of 
mathematics and natural philosophy. 

* It is remarkable that while he possessed such powers of calcu- 
lation, and was almost prodigal in the exercise of them, he was to 
the last degree solicitous about the metaphysics of every subject on 
which he undertook to write. We have seen a decisive instance of 
this tendency of his mind in his treatment of algebra considered as 
the science of pure time. So, again, in laying the foundation of 
his Calculus of Quaternions, we see him labouring to secure its 
stability by the most careful regard to the primary conceptions of 
time and space. Students of his Lectures on Quaternions have 
sometimes complained that he has claimed from them too much 
attention to the metaphysics of the subject, and has stopped them 
in their career of building up, in order that they might contem- 
plate afresh the plan of the structure. But this was in accordance 
with his views regarding the ascending scale of the subjects of 
human thought. To religion he gave the highest place and this 
not as a formality ; for his was a deeply reverential spirit. He 
assigned the next to metaphysics. To them he subordinated 
mathematics and poetry, and assigned the lowest place to physics 
and general literature. His studies in the department of meta- 
physics were extensive. After a thoughtful examination of 
Berkeley's writings, he professed himself a disciple of that 
philosopher, " with most cordial and delightful submission " ; 
not, indeed, assenting to every separate argument, but embracing 
his grand results ;* and in this attachment to Berkeley's theory we 
have reason to know that he was confirmed by his converse with 
Faraday, who in his own region of investigation had been led to 
the conclusion that forces, rather than material particles, were the 
ultimate objects of physical inquiry. His acquaintance with the 
German language enabled him to master the works of Kant. In 
the reasonings of that philosopher, he was the more ready to concur, 
as his own previous inquiries had already conducted him to several 
of Kant's views respecting the intuitions of time and space. . . . 

See, however, supra, p. 177. 



1865.] Dean Graves s Eloge. 223 

' His poetical compositions were the genuine outpourings of a 
noble heart and fervid imagination, characterized by a depth of 
thought and elevation of sentiment which compensated for occa- 
sional defects in artistic execution. These poetic efforts have an 
additional interest, as exemplifying in his own productions the 
connexion which he so strongly insisted on as existing between 
the highest provinces of science and the region of poetry in both 
of which he maintained that there was scope and demand for the 
exercise of the imaginative faculty. According to him, the modern 
geometry, which deals with the infinites and imaginaries of space, 
has its beauty and its fascination ; and he reckoned the happy 
daring of such geometers as Poncelet and Chasles as closely allied 
to poetry. We happen to know that this view of his, as communi- 
cated by him to the poet Wordsworth, was to the latter an entirely 
new revelation, and had the effect of raising his conception, which 
had before been unduly depreciatory, of the dignity both of science 
itself and of its most eminent votaries/ 

The testimony which follows in the Eloge to the faculty for 
business manifested by Hamilton in his discharge of the office 
of President of the Royal Irish Academy has been quoted in the 
second volume of this work, page 245, and need not be here 
repeated. Dean Graves then proceeds : 

' . . . A mathematician endowed with such original powers as 
Hamilton possessed might have been excused, if, yielding to the 
natural temptation of waiting for casual inspirations, he had 
carried on his labours in a desultory or unsystematic manner. 
To such temptations and no doubt he felt them he rose superior. 
He was, on the contrary, remarkable for the diligence and method 
with which he performed all his work. These qualities are evi- 
denced by the number, magnitude, and importance of his published 
works. There was no minute care, even in matters of typographi- 
cal nicety, which he disdained to expend upon them. And in his 
MS. books, carefully written, and with dates marking from day to 
day the progress of his scientific life, he recorded all his medita- 
tions, all the calculations through which he passed in his appar- 
ently fruitless, as well as in his most successful, researches. These 
volumes, many of them very large, and numbering about sixty, 



224 Life of Sir IVi I Ham Rowan Hamilton. [1865. 

have been deposited in the Library of Trinity College. They will 
supply to future historians of science the most precious materials 
illustrating the development of Hamilton's discoveries. They will 
exhibit, doubtless, germs of thought suggestive to others of new 
discoveries. They record a great commerdnm cpistolicum his 
correspondence with the most distinguished scientific men of his 
own age. Nay, more, they will be found to contain memoirs on a 
variety of subjects, complete in themselves, and carefully elaborated, 
but which he had abstained from publishing, either because they 
were unconnected with the greater works which he had in hand, 
or because he hoped to develop them more fully at some future 
time. It is to be hoped that they will yet see the light, and like 
the posthumous memoirs of Euler, inspire us with a feeling that 
their great author is still holding converse with us. It will be a 
satisfaction to the members of this Academy to be told that his 
Elements of Quaternions the work upon which he was engaged 
with most unceasing activity for the last two years is all but 
complete. I have reason to know that at no period of his life 
not even when he was in the prime of health and youthful vigour 
did he apply himself to his mathematical labours with more devoted 
diligence. Those who did not actually know how he was employed, 
or who had formed a false estimate of his character, might imagine 
him indolently reposing upon his laurels, or pursuing his studies 
in a desultory way. Such a conception of them would be the very 
opposite to the true one. His diligence of late was even excessive 
interfering with his sleep, his meals, his exercise, his social 
enjoyments. It was, I believe, fatally injurious to his health.' 

I will not refrain from adding the prognostication with which 
the Eloge concludes: 

* Believe me, gentlemen, the fame of SIR WILLIAM EOWAN 
HAMILTON, great as it was during his lifetime, will become yet 
greater when the world has been furnished with materials enabling it 
more perfectly to estimate the variety and richness of his endowments 
and the value of the services which he has rendered to science. 
Hifl reputation, eveii now, does not rest on the partiality of friends 
and country men. The learned men of all lands have a In 
declared him worthy of the highest honours which can be paid f<> 
intellectual eminence. This world- wide recognition, at the present 



1865.] Early Aspirations Fulfilled. 225 

time, of his genius and discoveries, affords us a sure pledge and 
earnest of the perpetuity of his reputation, and warrants us in 
regtarding his name as a glory which is not to pass away from the 
scientific and literary chaplet of Ireland. And in this fact and 
this anticipation we may thankfully and happily behold a full 
justification of his own early, and it might have been feared 
enthusiastic, aspirations of his deep and generous consciousness 
that he was intrusted with faculties and powers capable of achiev- 
ing in the noblest fields of thought a worthy fame both for himself 
and for his country.' 



It will be interesting here to recall, in connexion with the 
reference made by Dean Graves to the early aspirations of 
Hamilton, a passage in a journal written by him when a youth 
of twenty, in which he deliberately renews for himself words of 
aspiration to which he had given expression at a still earlier time. 
Recording the effect produced upon him by the contemplation of 
a great work like the Principia or the Mecanique Celeste, he goes 
on to say, in words combining modesty and strength : 

* When I see how much others have done, and contrast with it 
the little to which I have attained, the effect is painful but salu- 
tary. It seems practically to impress that eminence cannot be 
attained without exertion ; it teaches modesty of the most genuine 
kind, and in the most natural manner ; at the same time it acts as 
a powerful stimulus, and kindles the ardour of my aspiring after 
that fame which (as I once expressed it in a letter to my uncle) is 
the " meed which Genius and Industry when united have some- 
times been so fortunate as to obtain, with the world for their arena, 
and all time for the tribunal ; which has wedded to immortality 
some favoured names, and marked out some individuals as the 
instructors of mankind." ' (Vol. i. pp. 206-7.) 

A few words in a letter from Sir John Herschel to Lord 
Dunraven seem to me with great power to express a truth of high 
value in reference to Hamilton's enthusiasm. He writes : 

VOL. III. Q 



226 Life of Sir William Rowan Hamilton. [1865. 

From SIR JOHN F. W. HERSCHEL to the EARL OF DUNRAVEN. 

' COLLINGWOOD, December 19, 1865. 

* MY DEAR LORD DUNRAVEN A great many thanks to you for 
sending me Dean Grraves's Eloge on Sir William E. Hamilton, 
which is admirable ! in perfect good taste, and not a word more 
than was fully deserved. 

'He was indeed a most admirable person, and a most truly 
amiable and high-souled one. Nothing but so much greatness 
could have made so much enthusiasm only what was natural 
and nothing but so much enthusiasm could have carried him on 
to so much greatness.' 

An Eloge of high tone and ability was also pronounced on 
Hamilton, in the year 1866, by the Eev. Charles Pritchard, at that 
time President of the Eoyal Astronomical Society; and to the 
September number in the same year of the North British Revieir, 
Professor Tait contributed a lucid and powerful exposition of 
Hamilton's scientific achievements, interesting, from its style and 
substance, alike to the general reader and the student of science, 
and worthy of a disciple whom Hamilton in one of his latest notes 
to the Elements of Quaternions (p. 755, note) designated as 
'eminently fitted to carry on, happily and usefully, this new 
branch of mathematical science, and likely to become in it ... one 
of the chief successors to its inventor.' 

From these testimonies of eminent Men of Science I turn t '. >r 
the last word to the Poet Friend, known to Hamilton for five-and- 
thirty years, and at the end of fifteen additional years uttering in 
verse an affecting proof that the feeling of admiring and affec- 
tionate friendship was in his heart as deep and fresh as ever. 

I have already given to the reader one of two sonnets writ ion 
by Aubrey De Vere, in 1880, in memory of Hamilton.* Addressed | 

Supra, p. 159. 



1865.] Farewell Sonnet. 227 

to the spirit of his friend, it recalls attributes of his character 
which may here fitly be reproduced : 

' Thy soul translucent, yet thine insight keen, 
Thy heart's deep yearnings and perpetual youth ; 
Thy courtesy, thy reverence, and thy truth .' 

The sonnet which I reserved is the following retrospect of their 
friendship ; it sets before us, in succession, the two congenial com- 
panions united in the noblest aspirations ; the survivor, filled with 
4 beaming memories ' ; the departed, at the last in full secure 
possession of the truth he loved. Who can repress the hope that 
they will be re-united ? 

' Friend of past years, the holy and the blest, 
When all my day shone out, a long sunrise ; 
When aspirations seemed but sympathies, 
In such familiar nearness were they dressed ; 
When song, with swan-like plumes and starry crest, 
O'er-circled earth, and beat against the skies, 
And fearless Science raised her reverent eyes 
From heaven to heaven, that each its God confessed 
With homage ever widening ! Friend beloved ! 
From me those days are passed ; yet still, oh, still, 
This night my heart with influx strange they fill 
Of beaming memories from my vanished youth : 
On thee the temporal veil by Death removed 
Rests the great Vision of Eternal Truth ! 

* January 10, 1880.' 



Q2 



228 Life of Sir William Rowan Hamilton. 



CHAPTEK LII. 
(SUPPLEMENTARY.) 



A GATHERING OF FRAGMENTS. 

IN looking back over the letters and other documents in my hands,. 
I have come to the conclusion that some items, either accidentally 
overlooked or that found no natural place in the biography, have 
been omitted, which nevertheless are worthy of preservation as 
either expressing a thought of value or casting additional light 
upon some feature of Hamilton's character. 

Under the head last mentioned I give a passage from a frag- 
ment of a letter to Aubrey De Yore which is undated, but which 
must have been written in September, 1842 (see supra, vol. ii. 
pp. 393-4), soon after his perusal of Henry Taylor's Edwin //// 
Fair : 

* My copy of Edwin has been carried off by one of my nieces 
at Dunsinea, and I shall only at this moment say that I was 
particularly tickled by the sketch of Wulfstan the Wise, which, 
if I enjoyed the acquaintance of the author, I might have fancied 
to be a hit at me. The two volumes of Tennyson have only this 
instant reached me, and I have not opened them yet.' 

The sketch referred to is in Ed inn the Fair, act ii. scene 2, 
where Earl Leolf thus describes Wulfstan : 

1 This life, and all that it contains, to him 
Is but a tissue of illuminous dreams 
Ml I'd with book-wisdom, pictured thought, and love 
That on its own creations spends itself. 
All things he understands and nothing does. 



Thought (i)id Action . 229 

Profusely eloquent in copious praise 

Of action, he will talk to you as one 

Whose wisdom lay in dealings and transactions ; 

Yet so much action as might tie his shoe 

Cannot his will command ; himself alone 

By his own wisdom not a jot the gainer. 

Of silence and the hundred thousand tilings 

7 Tis better not to mention, he will speak, 

And still most wisely.' * 



There was doubtless some ground for applying such a descrip- 
tion to himself, however he may have laughingly felt that in sucli 
application it was a caricature, and we may remember his noting 
in early life (vol. i. p. 620) that he was conscious of having, like 
Coleridge, too much of the element of iraOoQ in his mental consti- 
tution ; but it is also true that, when there was an adequate call 
upon him for the exertion, he could display much practical ability. 
1 remember Dr. Humphrey Lloyd being struck by this after 
perusal of some early letters to Lord Adare, giving directions 
for the setting up of a dial letters, however, which were too 
elementary to print and I have already adduced testimony, which 
has been confirmed to me from many quarters, of his power 
promptly to apply in the way of practical decision his thorougli 
knowledge of the constitution and laws of the Eoyal Irish Aca- 
demy when acting as its President. A more adequate estimate 
of his character, drawn by his own hand, may be deduced from 
ihe following letter to Mrs. Wilde, whose poetic temperament 



* The continuation of the letter is worth giving in a note. Amusing in 
itself, it furnishes an additional instance of Hamilton's observing interest in 
the intellectual progress of his boys. * . . . My eldest boy is not quite so well 
<-is usual to-day ... I was much amused at the development of the opinions of 
his younger brother [Archibald Henry] a few mornings ago on the voluntary 
principle, which, when he found incapable of being pushed into a rigorous 
universality, he said that he should like to keep for himself, and to let the 
involuntary principle apply to other people. But William Edwin argued that 
it' the \oluntary principle were to be carried fully out, each man might com- 
plain of God for not giving him everything he wished to have more money, 
more rich fruits, and a grander and more beautiful earth.' 



230 Life of Sir William Rowan Hamilton. 

sometimes prompted her to give enthusiastic expression to her 
admiration of Hamilton's intellectual powers. 



'OBSERVATORY, February 11, 1858. 

* You know that whatever it may be at any time your pleasure 
to say, even it if be in praise of myself, I submit to it, from the 
profound conviction felt by me of your being an entirely truthful 
person. 

' Of course it is needless to say that I am not to be considered 
as adopting any expression which it would not have been modest 
enough in me to have first uttered, but to which I have (once or 
twice) listened, without remonstrance, from you for the reason 
mentioned above. 

1 It may not sound very consistent with any such professed 
humility on my part, if I say to you that, after having served for 
the Quaternions during fourteen years, and having (as America 
seems to think) won my Rachel to be my own by an intellectual 
marriage I now wish to wind up several scientific projects, from 
which those quaternions had for a long time diverted me ; and feel 
as if I were entering, or had already entered, on a new harvest of 
labour and reputation. As to Fame, if it have not been won or 
earned already, it is not likely that any future exertion will make 
it mine. 

' But as to the LABOUR ; that is a thing within everybody's 
power to judge of, even for himself. I have very long admired 
Ptolemy's description of his great astronomical Master, Hippar- 
chus, as avvp ^(XoTrovoc KCU $tAaA/j0?e " a labour-loving and 
truth-loving man." Be such my epitaph ! 

' Since I have presumed to translate (unnecessarily for you) a 
bit of Greek prose, let me try my hand at a small portion of Latin 
verse, of which I am reminded by the subject of this note. It 
occurs almost at the end of Juvenal's celebrated Tenth Satire : 
Johnson's very clever imitation of which (" Vanity of Human 
Wishes") is not within my reach just at present. The lines to- 
which I refer stand thus in the original from which I copy : or at 
least in a book given me long ago by Dr. Brinkley, who knew that 
I enjoyed the Classics, and which is dated, "Parisiis, 1528," and 
is printed throughout in the Italic character : 



Ideal of Character. 231 



' " Ut tamen et poscas aliquid, voveasque sacellis 
Exta, et candiduli divina tomacula porci, 
Orandum est, ut sit mens sana in corpore sano. 
Fortem posce animum, et mortis terrore carentem, 
Qui spatium extremum vitao inter raunera ponat 
Naturae, qui ferre queat quoscunque labores, 
Nesciat irasci, cupiat nihil, et potiores 
Herculis aerumnas credat, ssevosque labores, 
Et venere et caenis et plumis Sardanapali." 

4 My translation of a few minutes ago is as follows : 

' " Yet if thou must ask something of the gods, 
Pray that there be sound mind in a sound body. 
Ask a firm soul, one void of fear of death ; 
Which may rank life's last space with boons of Nature ; 
Be able to endure whatever labours ; 
Be never angry ; covet nothing ; count 
Better the pains and the stern tasks of Hercules, 
Than pleasures, feasts, and down of Sardanapalus." ' 

To the word * carentem,' in his transcript of the original passage, 
the writer adds the marginal note, * For " carentem " a Christian 
would have written " liberatum." 

It may be felt that in the two quotations here given, the Greek 
and the Latin, Hamilton recognised the ideal of a character strong 
and lofty, the possession of which he was conscious of having 
habitually aimed at, and of having in great measure attained. 

As another contribution to the estimate of his character, coming 
from his own hand, I add a short extract from a note to an in- 
timate friend, written at the close of 1852 : 

* ... Indeed I have much for which to seek forgiveness both 
from God and man but certainly I have never caused pain for 
any gratification of my own. . . .' 

Hamilton's tolerance of opposing opinions has been noted by 
Professor De Morgan (supra, p. 218) ; it was a tolerance not of 
mere good-nature or indifference. The following extract from a 
letter to Mrs. Wilde shows that in the subjects by which tolerance 
is most severely tested, religion and politics, he wisely recognised 



232 Life of Sir William Rowan Hamilton. 

the true ground for tolerance, namely, the large extent over which 
persons of good sense and good feeling were at one, however they 
might differ in religious creed or political party : 

' In religion and politics men can't, they ought not to think 
one view as good as another. We must not say, like Tooks in 
Domhey, it is " of no consequence at all " : but each may give his 
neighbour credit for being as sincere as himself. I think, how- 
ever, that with most people who do not vigorously shut their 
hearts against receiving impressions from others, and who mingle 
at all with the world, or even read pretty freely, there grows up 
gradually a feeling that for the most essential purposes of life, 
including thoughts and conduct, people who differ can yet sympa- 
thise with one another/ 

The sonnet which I here introduce would have been inserted 
after Hamilton's letter of January 30, 1835, to Aubrey De Yere, 
had it not been accidentally overlooked by me. It was written at 
the conclusion of some weeks spent at the Observatory in solitude, 
and on the point of his starting to pay a short visit to his wife and 
child, and Mrs. Bayly, at Bayly Farm, near Nenagh (see vol. ii. 
pp. 118, 120). Mrs. Bayly was the * Mother' of the last line. 

* Draws to its close a melancholy while, 
A long long night of absence ; from the sky 
Melts off the solid gloom ; pale phantoms fly, 
And soon the blushing dawn will brightly smile ; 
And like a man who many a weary mile 
Hath travelled lonely, if at length his eye 
Discern the wish'd for fountain, so feel I, 
A traveller near the Sources of the Nile. 
What I have pined for, what has been a power 
Guiding my steps through time as his through space, 
Imagined though unseen, will face to face 
Repay me soon for many a lonely hour : 
When I shall clasp, in a remember' d bower, 
Mother and wife and child in long embrace. 

OJWEHVATOBY, January 31, 1835.' 

This sonnet is a strong proof of the affectionate feeling which 
Hamilton never ceased to cherish towards his wife. Other 






In his own Family. 233 



instances have been given. Yet it cannot be denied that the whole 
course of their married life proved the j ustness of her early fore- 
boding, arising from a sense of weak health, both of body and 
mind, that she was not fitted to sustain the burden of duties pro- 
perly devolving upon a wife in her position. The following 
Idler I impart to the reader, principally because, written when 
lie had been more than twenty years her husband, it manifests the 
continued warmth of his conjugal affection, and the considerate 
1 houghtf ulness of a true paterfamilias, and, at the same time, 
.shows how he could even to herself playfully allude to her extreme 
shyness and retiredness. I remember hearing Dr. Lloyd say that 
he had been often at the Observatory, but had never seen Lady 
Hamilton. 

'LrvEEPOOL, September 27, 1854. 

' MY DEAREST HELEN The Association has just been adjourned 
to Glasgow, to which place I had moved, in the General Committee, 
that we should go next year. They got me up to make another 
speech, just now, proposing thanks to the foreigners, and I came 
out with flaming allusions to the war and our French allies. I 
mentioned my having learned mathematics chiefly from French 
books, but said that the authors of them, as I had never visited 
Paris, had appeared to me almost as abstract ideas. (You know 
that Lady Hamilton was once called by Dr. Robinson " an abstract 
idea.") But I amused the audience by turning to the fat little 
Abbe* Moigno, of whom Will has heard me speak in connexion with 
Cauchy, and who was sitting near me, and by saying, "For instance 
I had thought the Abbe here to be an abstract idea : but I think that 
you will all agree with me in considering him to be a very con- 
crete and a very pleasant body " ! Arch promises you that he 
\vill not become a "damp unpleasant body," if he can help it. 
But he ought to have written to you ere this. Indeed I know 
that he did begin a letter on the day of our arrival here, but mis- 
laid it among my papers. He seems to have enjoyed himself 
greatly. I am quite tired after all the meetings I have attended 
tn-day, and yet hope to attend another to-night, after resting a 
little at my lodgings. I cannot write to-day to anyone but you, 
but please to tell dear Moo [pet-name of his daughter] that I 



234 Life of Sir William Rowan Hamilton. 

received her letter, and that our lessons in French have heen of the 
greatest use to me. Give love to William also, and believe me to 
remain your very affectionate husband, W. E. Hamilton. Please 
write to me immediately, or let Will or Moo do so, directing as 
before. Arch and I will probably start for the Lakes on Saturday.' 

Hamilton's interest in children was not confined to his own. 
It was characteristic of him that he treated all children not only 
with tenderness but with respect. An extreme instance of this 
has been communicated to me by his surviving sister, from whose 
letter I transcribe it, rendered specially interesting, as it is, by the 
warmth of feeling with which it is narrated : 

' Before leaving Dublin for the neighbourhood of Blackrock I 
went out to spend a parting day at the Observatory, bringing with 
me a dear little charge, a sweet little girl of weak powers. Her 
mind had never opened at all, till she was put under my care. 
She had been in the Earlswood Institution for Idiots, but was no 
better, rather worse after a year. When she came to me she could 
not speak so as to be understood, could not hold a spoon or needle, 
and so on. This will just give you an idea of the little girl. It 
pleased God that I should see that she was not idiotic. She had 
memory, affection, the power of distinguishing colours and people, 
pity and sympathy, manifested when I showed her a picture in 
which were represented dead camels in the Desert, &c. ; and she 
won upon my heart, and she loved me beyond anything on earth, 
just because I first showed that I considered her worth trouble. 
" Lizzie CAN taught " was her delighted expression, when she suc- 
ceeded in buttoning her cuff for herself. She grew up a lovely 
girl, and Ellen, another young person with much talent and no 
deficiency, whom I trained from a child, and who is with me here, 
taught my poor Lizzie to sing sweetly, and she grew on improving 
miraculously till she was thirteen, when she died and went to 
Jesus, whom she loved. Still she was, as you can well suppose, 
far, far behind even common intellects in many respects. Well, I 
brought her to the Observatory : my dear brother and [his daughter] 
Helen knew her well at my lodging in Dublin. I left her in the 
drawing-room and went up with Helen to her room, and we st;r 
there talking, I knowing well that Lizzie would remain quietly 



if is Kindness all round. 235 

below till I went down : but after a time a message came up to me 
that Sir William sent to tell me " that he was now obliged to go 
to his library and Miss Lizzie would be alone;" he could not bear 
to be unpolite even to her. Of course I went down at once to ease 
///.s feelings, not the dear child's. He told me that he had played 
a game on the Icosian board with her, and " though of course I 
saw that she did not understand it," said he, "yet I assure you 
that many a duchess could not have carried it off so well. She 
had to me quite the air of one accustomed to Castle society, who 
considered that she condescended to oblige me." I almost felt tears 
rise as I listened to the truthful humble words of that Giant in 
Intellect, and thought of the pitiful sneers that poor nobodies 
would give at the idea of sitting down to play any game with 
poor Lizzie. He sent us into town on the car after dark, and he 
said to me, " I think her too interesting; I hope she will not attract 
anybody too deeply to her ; but I should fear it for her." Hia 
humility was to me the most wonderful thing about him/ 

She adds, as an anecdote illustrative of his truthfulness : 

' He gave a question to the boys of Lovell Edgeworth's school 
which they could not answer. Presently he found that the solution 
was impossible, and he at once avowed his mistake, for which 
prompt confession the boys gave him a tremendous cheer.' 

There must have been something frank and engaging in his 
way of doing what most examiners in a similar case would have 
done ; for I find the incident (which occurred in 1828) referred to 
in a letter of Maria Edgeworth. 

It will not be wondered at that Hamilton's kind feeling ex- 
tended to our dumb fellow-creatures. In a letter written when 
very young, from Drumcondra, he speaks of his spending the 
earliest hours of a summer morning in reading Shakespeare and 
playing with two kittens whom he had taken to bed with him, and 
his sister writes 

4 He was always fond of cats, and might often be seen writing 
some mathematical paper with a kitten or favourite cat on his shoulder 
playfully trying to catch the pen.' ' His politeness was almost a 



236 Life of Sir William Rowan Hamilton. 

rebuke to others. A young lady, who was living with me in 
Dublin at one time said, " I never saw so polite a gentleman as 
your brother ; I think he would almost bow to a cat; " and I 
reminded of her and amused him by repeating this to him one day, 
when accidentally he did tread upon the cat's paw, and turned 
round, and smiling said, "I was going to say, I beg your 
pardon." 

His feeling consideration for all living things around him 
drew towards him from them unbounded confidence. One instance 
of this made a deep impression on those who witnessed it, and 
indeed it was an occurrence fitted to excite and to excuse a some- 
what superstitious wonder. On a "Whitsunday morning, as he was 
reading prayers in the centre of his assembled household, a dove 
flew in through the open window and settled on his head ; it was 
undisturbed by Hamilton, who contiuued to read, and after an 
interval it peacefully flew out. 

It is not on record, I believe, that he ever killed or even struck 
in anger his mute fellow-creatures. A single exception, not un- 
characteristic, in regard to the latter statement, is related by his son 
as told by himself. Finding the greyhound, Smoke, one day tearing 
a Book of Common Prayer in the library, he thought it his duty, at 
some personal risk, to inflict upon him a serious chastisement. 
'This,' Mr. W. E. Hamilton says, ' was in his High Church 
days.' That he possessed abundant physical courage, and a 
strong sense of personal dignity, was habitually manifested by 
him : one illustration is the fact, that in his earlier days he 
challenged to a duel a member of the Royal Irish Academy, 
who, as he conceived, had impugned his honour or truth. His 
friend, Colonel Larcom, whom he engaged as his second, suc- 
ceeded in obtaining for him adequate verbal satisfaction. 

On the authority of Hamilton's intimate friend, Dr. Samuel 
O'Sullivan, the following report of an interchange of compliments 
between Wordsworth and Hamilton has been communicated to 
me: the circumstance that only an astronomer could have im- 
agined suoh a reply seems to authenticate the anecdote : 



Wordsworth and Hamilton. Philosophy. 237 

' When Wordsworth visited Hamilton at the Observatory [in 
1829] he took occasion to say, " I feel happy in a pleasure 
rarely enjoyed by me, that of being in the company of a man 
to whom I can look up." " If I," replied Hamilton, " am to look 
down on you, it is only as Lord llosse looks down in his telescope 
to see the stars of heaven reflected." 

A reply of Hamilton's, more in his style, calls for record. 
When asked whether he accepted, as expressing a truth, Locke's 
comparison of the state of the human mind at birth to a sheet of 
white paper, he said, ' Yes, but ruled paper ' : an answer pregnant 
witli much of his philosophy ; which in outline admits perhaps of 
no nobler adumbration than it has received from Wordsworth in 
the lines which form part of what he calls the Prospectus of his 
poom, ' The Recluse ' : 

1 while my voice proclaims 
How exquisitely the individual Mind 
(And the progressive powers perhaps no less 
Of the whole species) to the external World 
Is fitted : and how exquisitely, too, 
Theme this but little heard of among Men, 
The external World is fitted to the Mind ; 
And the creation (by no lower name 
Can it be called) which they with blended might 
Accomplish.' 

It will be remembered how emphatically Hamilton distin- 
guished in the mind the faculty of Intellect from the faculty of 
Faith. He felt, if I recall rightly what I heard from him, that 
no small part of the proof of the existence of (rod rested upon the 
fact that the great Idea fills, as no other idea can fill, the aspira- 
tion of Faith. In one of his Manuscript Books (I. 1864, pp. 
1512) Hamilton transcribes from pencilled notes of 1852, one 
upon Pantheism. Confessing that Coleridge's statement does not 
produce in me the satisfaction it appears to have given to Hamil- 
ton, I produce the note principally on account of the comparison, 
in regard to mathematics applied to metaphysical reasoning, of 
Coleridge and Kant.* 

* See vol. ii., pp. 137, H2. 



238 Life of Sir William Rowan Hamilton. 

' Pantheism, much more than Popery, appears to be the danger 
of our age. Coleridge confessed to me, and regretted the confes- 
sion indeed was needless, but the regret was interesting that he 
had never studied mathematics. Accordingly, in his philosophical 
writings, as a general rule, his mathematical illustrations have 
seemed to me to obscure the subject (while those of Kant throw 
light upon every point to which he applied mathematics). But I 
must make one signal exception, on this very subject of Pantheism, 
Coleridge's mathematical illustration of it appearing to me to be 
perfect. 

Let G (said he) stand for God, 

and W for the World : 

then theist and pantheist agree in asserting the formula, 

W- = 0; 

"The World, without Grod, is nothing." But, continued Cole- 
ridge, the pantheist inverts the formula, and says also, 

G- JF=0: 

or, " God, without the World, is nothing." Whereas (he said) the 
theist, on the contrary, asserts that 

G- W = G: 

or that " GOD, without the manifestation of himself, which he has 
been pleased to make in his created Universe, would still have 
been the same personal God" 

Turning now to Hamilton's own subject of Quaternions I 
transcribe a page from another of his Manuscript-Books (M. 1848. 
p. 73) which displays what was with him a favourite stanza of 
Beattie's Minstrel, and adds a comment. One can imagine how 
pleased was the inventor of Quaternions with its opening lines : 

From BEATTIE'S Minstrel. 

1 And Reason now through Number, Time, and Space, 
Darts the keen lustre of her serious eye ; 
And learns from facts compared the laws to trace, 
Whose long progression leads to Deity. 
Can mortal strength presume to soar so high ! 
Can mortal sight, so oft hedimmed with tears, 
Such glory bear ! for lo, the shadows fly 
From Nature's face ; confusion disappears, 
And Order charms the eyes, and harmony the ears.' 



J s Minstrel and Quaternions. 239 



' I had copied the foregoing extract from Beattie into another 
book, long before I thought of the Quaternions ; but it after- 
wards occurred to me that there was at least some distant analogy 
between the view of science taken in this passage, and that which 
I sought to embody in the two following lines of my own Sonnet 
entitled the Tctractys, which was one of the two that were jointly 
called by me " Recollections of Collingwood." * 

" And how the One of Time, of Space the Three, 
Might in the chain of Symbol girdled be." ' 

Early in 1866 Mr. W. E. Hamilton furnished me with some 
memoranda made by him respecting his father's habits of work 
and traits of character, the substance of which I here reproduce as 
the truthful notes of an observer possessed of special advantages. 

It will be remembered by the reader of this biography that 
Hamilton was accustomed, even up to the last year of his life, to 
work continuously in mathematical research or arithmetical calcu- 
lation for very many consecutive hours, the processes entered upon 
often requiring for completion such prolonged labour. To stop in 
the middle and rest he found impossible, for the mind would still 
work on, whether he laid down the pen or not, and he feared to 
lose the thread of argument or investigation. To continue to the 
end a task, in which good progress had been made, required, as he 
was convinced, support and stimulus for the brain, and this he 
-administered to himself in the injurious form of porter taken in 
small sips as he felt fatigued. The need thus experienced, con- 
nected as it was, with his disinclination to be disturbed at his work 
by regular meals, was, according to his son's testimony, the prin- 
cipal cause of his recourse to alcoholic stimulant, for which he 
admits that his father had besides a constitutional proclivity, as 
well as a disposition, arising from his genial nature, to conform 
to the prevailing custom of the time when he first entered into 
social life. 

Mr. Hamilton notes that his father was fond of teaching, 

* See vol. ii., p. 525. 



240 Life of Sir William Rowan Hamilton. 

partly from a genuine desire to impart knowledge to those whom 
he happened to address, who, however, were often incapable of 
receiving it, but not less} from a felt necessity of * throwing his 
mind into a didactic attitude,' in order that he might put the 
result of laborious analysis into the best form for communication. 
I transcribe here Mr. Hamilton's graphic description of the pro- 
cesses he observed : 

' It was his habit invariably before making a communication to 
the Royal Irish Academy to lecture an audience, fit, I hope, but 
few, consisting generally of myself and Thompson, the assistant, 
although the ladies were sometimes present. This was not a mere 
rehearsal, for the actual wording in the R. I. A. might be very 
different, but it was the throwing of his own mind into the didac- 
tic attitude, and the satisfying himself that he had done so. In 
fact he generally passed through three stages : 1. That of mathe- 
matical investigation ; 2. That of final polishing ; 3. That of 
throwing his own mind into the didactic attitude. I generally 
knew when 3 was coming, and anticipated it by bringing the 
black-board, taking care to have plenty of chalk ready. Well, 
then, he generally began by a few preliminary scribblings on the 
board, speaking to himself and rubbing out the chalk marks. 
Then followed a lecture, in which it was very curious to watch his 
tendency to digress. He would say, with a sudden start, " Yes 
wa it stop, I see another way of proving this ; let log tan 9 = &c., 
. . . hut, however, this will keep to resume, x = &c. Well, 
now we have killed off the first part of the subject." So that in 
44 lecturing " he suppressed the digressive tendency, but in conver- 
sation he yielded to it, and used often to say, " Well, what was I 
speaking of just before ? " " Of, &c. &e." Sir W., "no go still 
further back." It was very amusing to watch Thompson whose 
ideas moved slowly, and who could only go a very small way in the 
subject standing by, spectacled and owlish, and chiming in with 
an occasional " I see." Sometimes, however, my father threw a 
Parthian dart at him, such as "Just recapitulate the last six 
equations" ; when it generally happened that Spica Virginis or a 
Lyrso required immediate attention. I will add that the effort to 
throw his mind into the didactic attitude did not always succeed. 
Sometimes he had got to a certain stage in the lecture when some- 



Procrastination . 241 



new track of discovery suggested itself, and he made a pause 
noted sufficient to recover it afterwards and went on with the 
lecture : so that the didactic stage required a conscious effort in 
suppressing the inventive tendency for the time.' 

Under the head ' Procrastination,' Mr. Hamilton mentions of 
his father that * he was almost invariahly late for church, dinners, 
and public meetings of all kinds.' I think that Procrastination is not 
quite the right term for this short-coming. It arose, in my opinion, 
not from a weak habit of postponing what he had determined to 
do, but rather from a mistaken estimate of what might be accom- 
plished by him in the interval preceding an appointment. He lost 
count of time when absorbed in his own work, and, in consequence, 
engagements of inferior interest were deferred ; some, as it were, 
pushed forward so as to entail merely the lateness spoken of by 
his son ; some pushed off the line into an undefined future. It 
was thus miscalculation of time as an element to be applied to 
practical uses, not a failure of intention or even of will. He came 
late to church or dinner, but still he arrived. He answered a letter 
a month, or a year, after he received it, but he answered it ; some- 
times, indeed, omitting to post his answer : and as, speaking 
broadly, he forgot nothing, he only wanted a stretching out of the 
twenty-four hours, or a boundless extension of life, to fulfil all his 
obligations. He was certainly open to the taunt, friendly or hostile, 
that, however great a master he was of ' pure time,' he was no 
adept in the management of sublunary time of the time we have 
to deal with in this practical world of ours. And yet, after all, it 
may be asked, Is not the deficiency thus commented on the almost 
inseparable shortcoming as to minor activities of a great mind 
habitually employed in doing great things?* But the reader 
must not be deprived of his son's dramatic exemplification of work 
thus miscalculated with respect to time. This instance is, indeed, 
but a miniature of Hamilton's repeatedly unfulfilled prognostica- 
tions of the completion and publication of his books. 

* Sec as to "Wordsworth, vol. i., p. >"> 
VOL. III. R 



242 Life of Sir William Rowan Hamilton. 

' Mrs. Comerford could give instances of procrastination. She 
was much at the Observatory, and often used to go into town for 
marketing. One of Sir William's Laws of the Medes and Persians 
was " No servant, workman, postman, baker, College tradesman, 
or other person, shall go into Dublin or leave the Observatory 
without giving previous notice to me." Another, "Any person 
wanting any large sum (e.g. 5) must give ample notice (say a 
fortnight's) beforehand : anyone going into town for marketing 
and requiring any small sums shall give notice on the previous 
day." Mrs. C. sometimes forgot these rules, and one day was 
just going off at, say, 10 A.M. Dialogue : Mrs. C. " I am just 
going into town, uncle, and I want seven shillings to get tea, &c. 
Have you anything for town ? " Sir "W. " Oh, why didn't you 
tell me last night ? I could have had a package ready : you can 
come up in about an hour." 11 A.M. Mrs. C. "Well, uncle, are 
you ready ? It's 11 o'clock ; I wish you would let me off." Sir W. 
" I think I'll wait to write some letters." Mrs. C. " Shall I send 
the car round." Sir W. " Certainly not." Mrs. C. " The mare 
will be drenched." Sir W. " You can come up in about half an 
hour." 11.30 A.M., Mrs. C., knocking. " Well, uncle ! "-Sir W., 
with a groan. " What is it?" Mrs. C. "It's teeming rain, the 
mare had better go round." Sir W. (second groan). "Very well, 
I will wait for the evening postman " (4 P.M.). 4 P.M., postman 
comes. Mrs. C. " Well, uncle, I'll be late for shopping." Sir W. 
(cheerfully and abstractedly to himself). " Well, at last I've con- 
quered that." Sir W., to me. " I've got the second elimination in 
a very compact form. Now, you see 0," &c. Mrs. C. " Well 
uncle," &c. Sir W. " Wait a moment." " WeU then (to me) I 
see a very simple way stop, don't speak for a moment." Exit 
Mrs. C., whispering to me, " Get uncle to let me go or the shops will 
be shut, and our candles are out." 6 P.M., Mrs. C. (almost crying), 
" Uncle, the shops will be closed if you don't let me off." Sir W. 
" ft' 1 L = Well, I'll not mind writing any letters. I'll not keep 
you." Mrs. C. " But you did not give me the seven shillings. "- 
Sir W. " You should have told me before. However, mind I'm 
not keeping you." Exit Mrs. C., in despair.' 

Mr. Hamilton does not report much in connexion with religion, 
but states that his father did not hold the verbal inspiration of 



Hamilton* s Numbers. 243 

Scripture, and that he used to express a belief in the semi-inspira- 
tion of Milton : that, attaching high importance to the doctrine of 
the Resurrection of the Body, he believed it to be a necessary but 
sufficient condition that the risen body should contain identically 
some material particle or particles of the old body, however enter- 
ing into new chemical or organic combinations. He did not 
hold with the revivalists that it was the duty of laymen to try to 
convert others, but he considered it every Christian's duty, when 
infidel opinions were delivered in his hearing, to assert his own 
belief; a duty at least on one occasion publicly fulfilled by himself. 

I may conclude this Chapter of Fragments by recording what 
I had the pleasure of hearing this summer (1887) from Professor 
Sylvester : that he had lately found, in Hamilton's Memoirs con- 
nected with Mr. Jerrard's Eesearches as to the solution of equa- 
tions of the fifth degree, a discovery of Hamilton which had 
unaccountably failed to attract from succeeding mathematicians 
the attention it deserved. Professor Sylvester has made farther 
development of this discovery, which he has communicated to 
Crelle's Journal, and to the Transactions of the Royal Society, in 
a memoir entitled On Hamilton's Numbers, and he mentioned to 
me as a remarkable fact, that the first step in this direction was 
made by Bring in the year 1786 ; the next, and most important, 
by Hamilton, fifty years after, in 1836 ; and the extension by 
himself at the end of another fifty years, in 1886. This is an 
example of a published result which, having been long neglected, 
has been at last recognised as valuable, and turned to account. 
I cannot but hope that among Hamilton's unpublished manu- 
scripts may be found other mathematical results worthy of being 
given to the public, and likely to exercise profitably the attention 
of mathematicians. 



R 2 



IN the ensuing pages the reader is presented with a copious selection 
from the very active correspondence which commenced in 1841 and was 
carried on, though with long intermissions, up to the last year of Sir W. 
R. Hamilton's life, between him and Professor De Morgan. It will be 
found to be of interest not only to the scientific but also to the general 
reader. Had it been exclusively mathematical, I should have left it to 
take its place in his purely scientific correspondence, of which there are 
large remains. Its title to be thus connected with the biography arises 
from the fact that in it, upon subjects outside mathematics, the wit of 
De Morgan and the geniality of Hamilton, the mutual confidence, the 
comprehensive sympathies, and the honest divergencies of the two friends 
found free and characteristic expression ; while, at the same time, so often 
does the scientific element suggest the non-scientific, that it would be 
impossible without injury to both to part one from the other. 

When about to transmit to me the correspondence, Professor 
De Morgan, in terms which demand my grateful acknowledgment, wrote 
to me as follows : 

< 91, ADELAIDE ROAD, N. W., July 17, 1866. 

' I have a letter from young Hamilton, who is off to the New World 
before I could answer, asking me to transmit to you Rowan Hamilton's 
letters to and from me for selection. This I shall be very glad to do ; 
but you will find them an unsorted mass. If, however, you are going to 
set to work at a biography, you will find, as I have done, that a man must 
sort for himself, let who will try to precede him. Secondly, with refe- 
rence to his request that I would pencil anything I want to be omitted, I 
shall do no such thing. I will trust to your general intention not to 
insert anything that would give pain to the living or offence to the 
nearest of kin to the dead. I think it unlikely that I should sti ike- 
any thing out of my own, unless something should occur which tlu-iv 
could be no doubt about, and this you would be sure to settle without 
me.' 

In making the following selection it has been my study to obsi-rvi- 
the rule here laid down. 

A few of the earlier letters have been already inserted in the second 
volume of this work, but it has been thought expedient to reprint them 
in their proper places in the correspondence. 



CORRESPONDENCE 



BETWEEN 



SIR WILLIAM ROWAN HAMILTON AND PROFESSOR 
AUGUSTUS DE MORGAN, 



From PROFESSOR DE MORGAN to SIR W. E. HAMILTON. 

'69, GOWER-STREET [LONDON], May 8, 1841. 

' MY DEAR SIR WILLIAM I hardly know whether you remem- 
ber that we made a little personal acquaintance, some twelve years 
ago, when you were in London. 

* I take this opportunity of leaving my card with you in the 
accompanying form* by the post. 

* I shall be very glad to see the Theory of Triplets pointed at in 
your Paper on Algebra; time-triplets or space-triplets, I don't 
care which. 

' In the meantime, I remain yours very faithfully, 

'A. DE MORGAN/ 

From SIR W. E. HAMILTON to PROFESSOR DE MORGAN. 

' OBSERVATORY OF TRINITY COLLEGE, DUBLIN, 
'Jl%12, 1841. 

' MY DEAR SIR I have within these few minutes received 
your Paper On the Foundation of Algebra, and have hastily cast 

* Professor De Morgan's first Paper On the Foundation of Algebra, published 
in vol. vn., part ii., of the Cambridge Philosophical Transactions. See Preface 
to Lectures on Quaternions, p. (41). 



246 Correspondence between Sir W. Rowan Hamilton and 

my eye over it, intending to read and think about it afterwards. 
The handsome manner in which you have, there and elsewhere, 
expressed yourself respecting me, would render it impossible for 
me to be offended at the expression of some difference of opinion, 
even if such difference should turn out to be grave and irremovable. 
I am very sensible that, besides general dulness and heaviness of 
style, there is too much obscurity, in my Essay on Algebra as the 
Science of Pure Time, and one thing I am, and was, prepared to 
admit, nay, if it had seemed needful, to contend for, that Algebra 
does not require, for its foundation as a Science, any knowledge or 
conception of the actual succession of events, or of the relation 
of cause and effect; continuous progression appeared, and still 
appears, to me sufficient; but this, I thought and think, is 
the essential element in the conception of what I call pure 
time. Whether I am right in using this last form of expression 
is in a great degree, nay, almost wholly, a metaphysical question, 
in deciding which for myself I confess that I have been much 
influenced by study of Kant's " Pure Reason," and let me own 
that I am not prepared to decide, with you, that it is possible for 
a human mind to " imagine a given length to be instantaneously 
generated, no one portion of it coming into the thoughts before or 
after another," in opposition to the teaching of Kant, which seems 
to me to be confirmed by my own consciousness, that "we can 
think to ourselves no line, without drawing it in thought" (wir 
konnen uns keine Linie denken, ohne sie in Gedanken zu ziehen). 
Kant adds, "nor even" (we cannot even form the thought of) 
" time itself, except by drawing a straight line, to serve as its external 
construction, giving, however, attention only to the process of that 
synthesis of the manifold whereby we successively determine the 
inner sense, and thereby attending only to the successiveness of this 
determination" (und selbst die Zeit nicht [denken konnen], ohne, 
in dem wir im Ziehen einer geraden Linie, die die ausserlich figur- 
liche Vorstellung der Zeit seyn soil, bloss auf die Handlung der Syn- 
thesis des Mannigfaltigen, dadurch wir den inneren Sinn successiv 
bestimmen, und dadurch auf die Succession dieser Bestimmung in 
demselben, Acht haben). I cannot say whether this passage was 
in my recollection when I was drawing up my Paper on Algebra, 
but I remember that a similar train of thought prevented me from 
yielding to the suggestions of some friends, who were of opinion 



Professor A ugu si us De Morga n. 247 

that without much impairing the statement of my own view I 
should be likely to escape much opposition if I contented myself 
with speaking of continuous SUCCESSION, or progression, without 
introducing the jealousy- exciting name of Time. At all events, 
to show how naturally my view of algebra falls in with Kantian 
views of mind, though I am not aware that it presented itself to 
Kant himself, or to any of his commentators or disciples (with 
whose writings indeed I am but slightly acquainted), before the 
publication of my Essay, I may mention that in a work labelled 
Kant's Mctaphysic of Ethics, translated by J. W. Sample (octavo* 
Edinburgh, 1836), the dependence of not only "Arithmetic" 
but "Algebra," "the Calculus," &c., on the intuition of time, 
is familiarly spoken of, though without any reference that I have 
observed to my remarks, published the year before. 

1 But you are not to consider me as sworn to adopt the words 
of Kant ; and, so far as authority goes, I gladly own that I concede 
great weight to yours, on any question respecting the Metaphysics 
of Mathematics. I remember with great pleasure my introduction 
to you in London, and am glad that you too have me in remem- 
brance ; your works, through presentation or purchase, have for 
the most part reached, and interested me. 

' And I am, my dear sir, very truly yours, 
* WILLIAM R. HAMILTON.* 

'P. S. May 14. As to Triplets, I must acknowledge that 
though I fancied myself at one time to be in possession of some- 
thing worth publishing about them, I never could resolve the 
problem which you have justly signalised as the most important in 
this branch of (future) Algebra ; to assign two symbols Q and w, 
such that the one symbolical equation 

a + bQ, + Ch) = ai + b^O, -f Cjw 
shall give the three equations 

a = !, b = bi, c = c t . 

But, if my view of algebra be just, it must be possible, some way or 
other, to introduce not only triplets but polyplets, so as in some 
sense to satisfy the symbolical equation 

a = (a i, a->, a n ] ; 



248 Correspondence between Sir W. Rowan Hamilton and 

a being here one symbol, as indication of one (complex) thought ; 

and <?i, S , (t n denoting n real numbers, positive or negative, 

that is, in other words, n dates, in the chronological sense of the 
word, only excluding outward marks and measures, and the notion 
of cause and effect.' 

From A. DE MORGAN to SIR W. B. HAMILTON. 

' January 3, 1842. 

* I have undertaken for the Astronomical Society to look at 
Hansen and yourself, with a view to see whether his mode of 
using the elements of an orbit has been forestalled by you. 

* The only Paper I know of in which you have treated the 
passage from the general equations to the mode of developing 
them is the " General Method in Mechanics," Philosophical Trans- 
actions, 1834. Is there anything else which I ought to look at of 
yours on the same subject ? if so, will you oblige me with a refe- 
rence to the locus in quo. 

'But should it so happen that you have studied Hansen's 
various Papers, and can tell me anything yourself on the point 
without new trouble, I shall be very glad indeed to hear from 
yourself what you think of any connexion of methods which exists 
between the two yourself and Hansen. 

* I duly received your note ; I have since sent an a priori defini- 
tion of A* to the Cambridge Philosophical Society, which renders 
the new algebra all explicable a priori: thus f dv ~ 1 is made a 
creature of definition, not of subsequent interpretation. It is of 
course the a priori introduction of what answers to the logarithm 
of a number, which I call the logometer of a line given in magni- 
tude and direction. 

' Def. 1. On OX lay down Nap. log. of the length of A; on OY 
lay down the arc (rad. unity) of the angle A OX: the line having 
these projections is the logometer* of A. 

' Def. 2. By A B is meant the line whose logometer is 

B x logom. A. 

* Every line has of course an infinite number of logometers, according as we 
use 0, 02*, &c., for the angle. 

[The reader can easily supply for himself the ordinary figure of rectangular 
coordinates here referred to. Compare Transactions, Camb. Phil. Soo., \il. 
vii., part Hi., p. 292.] 



Professor Augustus De Morgan. 249- 

* For example ; one value of - } ^ - is TT ; as thus 

v/-l 

_ j ( log. of its length 0, 

j its angle TT ; 

logom. (-1) is line TT measured on OY: 
Division by <f- 1 is turning a line backwards through a right angle ; 

= TT measured on OX, whose symbol is simply TT. 
v ~ 1 

6 There is nothing now which ever gives me any thought or care 
in algebra except divergent series, which I cannot follow the French 
in rejecting. But I feel confident that in time the full import of 
these things will appear.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

* OBSERVATORY, January 28, 1842. 

* I was on the very point of leaving home for a short time when 
your note reached me, nor, though I have returned, can I yet say 
more in reply than that as you do not mention my Second essay in 
the Philosophical Transactions, it is barely possible you may not 
have seen it. Hoping to write soon again.' 

From A. DE MORGAN to SIR W. R. HAMILTON. 

'January 31, 1842. 

* Many thanks for your note. Since writing I have looked at 
your first [Essay on Dynamics] : the second I remember, now you 
mention it. 

* We have awarded the medal to Hansen, and it will be given 
on Friday week. His line and yours are different ; he has a 
curious way of making the time t into something else, and making 
this serve for variation of elements. Of course it only sets one 
thing right longitude for instance ; and he then corrects his new t 
to put the radius vector right. Airy calls it original, and there is 
no doubt he is a sagacious and painstaking man. 

'Now if anything should strike you as to your rapport with 
him, it will be very good for our President's Address on Friday 
week ; into which it can be incorporated. . . But you see that I 
need not ask you to trouble yourself as I have done, if your paths 
had been nearer together, to avoid mistakes.' 



250 Correspondence between Sir W. Rowan Hamilton and 



From SIR W. E. HAMILTON to A. DE MORGAN. 

* OBSERVATORY, February 8, 1842. 

'Though hard at work just now upon another subject, I write 
a line to mention that my recollection of the nature of Hansen's 
researches entirely agrees with yours; and that though I regret 
not to have been able to give to them much attention, yet I know 
enough of them to feel sure that they well deserve a medal. 

4 My own investigations in Dynamics lay in quite a different 
direction ; they conducted me to a system of rigorous and general 
expressions for the integrals of the differential equations of motion 
of any system of material points, attracting or repelling each other 
according to any function of the distance. 

* I understand that Jacobi considered my results important, and 
believe that he has put them under a still more general form. 
Whether any allusion to them could be appropriately made on the 
occasion you refer to will be better judged of by others, who are 
on the spot, and know what is to be said about Hanseii. Certainly 
there is no analogy of method, although the subjects are connected. 
My Introductions, in the Philosophical Transactions, contain a sum- 
mary of what I have done, or attempted, in the matter. 

* I am now writing out for the press, with some additions and 
illustrations, a Paper on Fluctuating Functions, which I communi- 
cated to the Academy in the summer of 1840, and which I have 
some hope will interest you. At all events I shall request your 
acceptance of a copy, when it is printed. It contains a new proof 
of Fourier's theorem, with generalised forms of that and other 
expressions for arbitrary functions, or parts of them.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 October 11, 1844. 

* I hope this will find you better in health than Graves repre- 
sented you when I saw him last. The Cambridge Philosophical 
Society asks authors for abstracts now ; accordingly I send you mi 
abstract of a Paper which I have just sent down to Cambri<! 
You will see that you are concerned in the concoction, and tlint 
though you will not triplicise, yet " numero deus impure gaudet " 
may find followers. 



Professor Augustus DC Morgan. 25 1 

' The first-mentioned system is rigorous common algebra for 
every plane passing through the axis of x. When two lines are 
not in one plane with that axis, still AB = BA ; but when three 
lines are in different places, then A (BC) is not (AB) C. 

1 Trusting to hear better accounts of your health, I remain, &c. 

* If ever you looked, as perhaps you did for a while, to resolve 

(a z + b z + c 2 ) (a* + b' 2 + c 2 ) = A* + B 2 + C\ 

you were trying the following : 

* " To find three points on a sphere each of which is opposite to 
both of the other two." ' 

'Abstract of a MEMOIR ON TIIIPLE ALGEBRA. 

* The extensions which have successively been made in alge- 
braical interpretation have been consequences of efforts to interpret 
symbols which presented themselves as necessary parts of the alge- 
braical language which is suggested by arithmetic. The now well- 
known signification of a + b ^/ - 1 did not yield any new imaginary 
or unexplained quantities : and accordingly no effort (within the 
author's knowledge) was made to produce an algebra which should 
require three dimensions of space for its interpretation, until Sir 
William Eowan Hamilton wrote a Paper (the first part of which 
was published in the Philosophical Magazine before the present one 
was begun) on a system of " quaternions." This system, as the 
name imports, involves four distinct species of units, one of which 
may by analogy be called real, the others being imaginaries, as 
distinct from one another as the imaginary of ordinary algebra is 
from the real. These imaginaries are not deductions, but inven- 
tions : their laws of action on each other are assigned. This idea 
Mr. De Morgan desires to acknowledge as entirely borrowed from 
Sir William Hamilton. 

Sir W. Hamilton has rejected the idea of producing a triple 
algebra, apparently on account of the impossibility of forming one 
in which such a symbol as a% + br} + c% represents a line of the 
length </ (a* + b* + c 2 ). Mr. De Morgan does not admit the ne- 
cessity of having a symmetrical function of a, b, c ; and, throwing 
away this stipulation, points out a variety of triple systems, par- 
tially or wholly interpreted. 

' Sir W. Hamilton's quaternion algebra is not entirely the 



252 Correspondence between Sir W. Rowan Hamilton and 

same in its symbolical rules as the ordinary algebra ; differing in 
that the equation AB = BA is discarded, and AB = - BA supplies 
its place. 

' Those of Mr. De Morgan's systems which are imperfect all give 
AB = BA, but none of them (the imperfect ones) give A (BC) = 
(AB) (7, except in particular cases. 

' Mr. De Morgan gives systems of triple algebra which he dis- 
tinguishes into quadratic, cubic, and biquadratic, according as the 
invented imaginary units represent square roots, cube roots, or 
fourth roots, of the negative real unit. It would not be easy in 
an abstract to give any account of them : but among them are 
found 

* 1. An imperfect quadratic system strongly resembling the 
common double algebra, and which would, but for its imperfect 
character, be at once recognised as the proper and natural extension 
of the interpretation of imaginary quantities to three dimensions 
of space : the ultimate symbol for a line is 

J(cos0+sin0 y^T). 

* 2. An imperfect quadratic system, very like the former one, 
except in having a peculiar inversion in the operation of multipli- 
cation, and a somewhat remarkable mode of representing what 
would by analogy be called arithmetical multipliers. 

* 3. A perfect quadratic system, the interpretation of which has 
considerable resemblance to that of the first-mentioned system, 
and is completely attainable, though not of great interest. 

'4. Three perfect cubic systems, each irreconcilable with the 
others, though closely connected with them. Each system presents 
a triple trigonometry, the cosine and two sines of which are each a 
function of two angles; but these can be easily expressed as 
functions of common circular and hyperbolic sines and cosines. 

* The interpretations of these systems are very imperfect, and 
appear to present great difficulty ; but their symbolical character 
is unimpeachable. 

*5. A perfect biquadratic system, which is of a redundant 
character ; that is, its fundamental form represents a line drawn 
in space from a given origin, with a symbol to spare, wlm-h may 
represent the time of drawing it, its density, its tendency t 
given position, &o., at pleasure. 



Professor A ugustus De Morga )i. 253 



1 Many interpretations are attainable, but Mr. De Morgan does 
not pretend to say he knows the one which ought to be adopted. 
It is singular that every attempt to reduce this algebra, by assign- 
ing a condition among the subsidiary symbols of its fundamental 
form, leads to an imperfect algebra. The system first mentioned 
in this abstract is one such result, and fails in its rules of multipli- 
cation, as before mentioned. Another is obtained, which is perfect 
as to its rules of multiplication, but fails in rules of addition.' 

From SIR W. E.. HAMILTON to A. DE MORGAN. 

1 OBSERVATORY, TRINITY COLLEGE, DUBLIN, 
'December 9, 1844. 

* I must have appeared discourteous in not sooner acknowledging 
your very kind letter, written to me about two months ago ; yet trust 
that you have not been displeased with the terms in which I noticed 
it in the supplementary number, for this month, of the Philosophical 
Magazine. Immediately after your letter arrived came an old 
friend [John T. Graves] to visit me ; and if you have ever been 
afflicted with the disease of procrastination, you must know that a 
slight cause, preventing immediate action, may be sufficient to 
produce a long delay. 

' I say nothing of slight attacks of ill-health, respecting which 
you are good enough to inquire. 

* You must not say that I refuse to triplicise : I had made 
a great number of attempts in that way, of some of which I have 
quite lately, and subsequently to communications from you and other 
friends, begun to think that they might have been worth pursuing. 
But it is too late for me to claim any merit in the matter, or even 
infancy myself entitled to any; I was prepossessed with an objec- 
tion which is gradually melting away, and shut my eyes against 
some things which now seem obvious. 

1 Yet I cannot altogether regret this result, even as respects 
myself, if I do not err in my estimate of the prospects which may 
belong to my own quaternion-theory, when it shall come to be 
taken up by abler hands than mine. It will surprise me, I confess, 
if either your theory, or any other person's, of pure triplets shall be 
found to surpass that which I have been led to perceive, as included 
in my theory of quaternions, on all, or most of, the three following 
points : 



254 Correspondence between Sir W. Rowan Hamilton and 

{ 1st. Algebraical simplicity ; . . . analogy to ordinary algebra, 
as to the rules of addition and multiplication (the commutative 
property excepted) ; 

'2nd. Geometrical simplicity ; . . . ease of construction ; the rule 
of the diagonal ; and, above all, symmetricity of space, no one direc- 
tion being eminent ; 

* 3rd. Deter minaten ess of division; ... a quotient being never 
indeterminate or impossible unless the constituents of the divisor 
all vanish. 

' Of all these assumed requisites, or things aimed at by me (and 
I admit that I aimed at others), what now appears to me most my 
own is the SYMMETRICALNESS or SPACE in my system. If you have 
succeeded in representing this with pure triplets, eris mihi maynm 
Apollo. My real is the representative of a sort of fourth dimension, 
inclined equally to all lines in space. 

* P. S. May I mention as an example of the working of my 
system, what I mentioned last month to the Royal Irish Academy, 
namely, a solution of the (certainly easy) problem respecting the 
resultant of any number of forces, applied at a common point on 
the last side of a closed polygon ? 

= 0i + 2 + 3 + . . . + n ; .*. 2 = tV + 2 2 + 3 2 + . . . + r, t 2 
+ (0i 2 + 2 0i) + (0i 3 + 03 0i) + (02 0s + 03 02) + &c. ; 



but 0i* = minus the square of the length of 0i ; and 0i 2 + r 2 n = 
minus the double of the product of the lengths of 0i and 2 x the 
cosine of their inclination ; .*. &c. 

* I also gave a theorem respecting the composition of finite 
rotations, deduced with great ease from my general method. An 
impossible line, as a tangent to a sphere from an internal point, is 
often indicated, in my theory, by having & posit ice square. 

' 2nd P. S. To-day, in looking over my old notes respecting 
triplets, I observed a recent one, dated the 25th of last September, 
written while I was thinking of going or writing to York, and 
intended as an argument against triplets; namely, that if we 
assumed t and j to be two such square roots of negative unity as to 
have their product = positive unity, but to be unconnected with 
each other by any linear relation, we should have 

j " ///" -\ jz" - (x + iy +jx) (x + iy +jz), 



Professor Augustus De Morgan. 255 

where 

x" = xx -(y-z) (y - z), y" = xy' + yx, z" = xz' + zjf ; 

my objection to which system, at the time, was this, that it 
gave (i+j) (% + iy + jz') = (*+/)#'> and. therefore caused the 
quotients (i + j)~\ (i + /)~ 1 > (*' +j)~ l jy to he all absurd, or impos- 
sible. (Of course I knew that in the ordinary theory of imaginaries 
the assumed equations relative to i and/ gave i + j = ; but the 
spirit of the attempt in question led me to exclude that relation as 
being linear). But looking at the subject again to-day, in the 
light of a very interesting communication, received about a fort- 
night ago, from my friend John Graves (whom I authorised to 
give you a sketch of my views about quaternions, when writing 
first to him on that subject, in the postscript of my lately printed 
letter), I seem to see that it might not be wholly labour lost to 
de^elope the consequences of these equations of multiplication ; 
and certainly not difficult to construct them. 

' Through each factor-point, xyz, and x' y f z', and through the 
product-point x"y" z", conceive an elliptic cylinder to pass, of the 
form x 1 + (y - s) 2 = ju 2 = square of what may be called the modulus ; 
then [/' = fj. ILL ; the projections on either of the circular sections or 
on either of the planes of xy and xz (made by lines parallel to the 
indefinite axis of the cylinder) obey Mr. Warren's rule; and if 
we make x = r cos $, y = r sin $ cos ^/, z = r sin sin i//, we shall 
have the equation tan sin (i//" - i/) = tan $' sin (\j/ - i//"). 

' The modulus is the semi- axis major of the ellipse perpendicu- 
lar to the indefinite axis of the cylinders. 

* 3rd P. S. I would not, if I could, prevent the study of pure 
triplets ; I feel for them something of the affection of a first love, 
for I was rather intimate with them once, though I have since been 
drawn away by what I thought superior attractions.' 

From A. DE MORGAN to SIR W. B,. HAMILTON. 

4 7, CAMDEN--STREET, CAMDEN Toww, 
December 16, 1844. 

* I am much obliged by your note. I have carefully abstained 
from your quaternions till now, and shall abstain till I have 
corrected my proof, which is now before me. 

* We are clearly on different tracks, and both necessary ones. 



256 Correspondence between Sir W. Rowan Hamilton and 

You are all for interpretation, and prepared to take new symbolic 
rules to get it ; my object is strictly to keep the symbolic rules of 
common algebra, and to let meaning come if it will. 

' I strongly suspect that you have the right sow by the ear, and 
that easy interpretation requires that the run of xy yz zx should be 
different from that of yx xz zy. 

6 My systems are not well interpreted, except by dropping (not 
changing) a symbolical rule. What may come of it I don't know ; 
I had but a fortnight at it before a slight attack of illness came 
on, and when I was able to work again, my lectures claimed me, 
and I have not seen the subject again till this proof arrived. 

' But I now see that your system is triple. You may say with 
Lord Byron 

" That you devoutly wished the three were four 
On purpose to believe so much the more." 

I have a triple system just like it, with an additional undetermined 
agent. Graves says your real quantity is only a kind of agent 
[or sub-agent?*] upon the multiplications, &c. : I suspect my 
biquadratic triple system has some very strong affinities with your 
quaternions, though the positive interpretation will be very diffe- 
rent. 

* However this may be, I am convinced that any system which 
does business with rotations cannot have xy = yx. 

' Graves gave me some extracts from your letter now published. 
His head ran on the transformations of sums of squares into other 
forms. He never dropped a hint about imagining imaginaries. 
On such little things do our thoughts depend. I do believe that 
had he said no more than " Hamilton makes his imaginary quanti- 
ties," I should have got what I wanted. 

'The system described by you with modulus 

+ (b-c)* 

is one of mine which I do not print, preferring one with a trifling 
variation which gives a? + (b + c) 3 ij = - 1 . 

' I had described this system to Graves, as a person describes 
in conversation.' 



* Writing nearly effaced. 






Professor A u gust us De Morgan . 257 



From the SAME to the SAME. 

' 7, CAMDEN-STEEET, CAMDEX TOWN, 
1 December 30, 1844. 

* I send you the proof of my Paper (done with ; you need not 
return it), which will perhaps give you my view of the matter two 
or three weeks earlier than I could otherwise communicate it. 
When clean and presentable copies arrive I hope to supersede this 
ragged affair. I have no hope of being able to think much about 
the matter before July. I hope I shall then be able to satisfy 
myself about the proper interpretations. 

* I look to what I have called the redundant biquadratic system* 
as giving the best chance. 

' Have you ever considered the vexata quccstio of fractional 
differential co-efficients ? We shall never arrive at the full com- 
prehension of integration until we know what 

f*\* 

. <bx 

\dx) 

means for all values of n. When divergent series are understood, 
and 



is as well known in all cases as when a = 1, /3 = 0, we shall be in 
a very good state. 

' Here is a quirk of analysis which was given in a Cambridge 
examination paper 

x rt (sin x sin 2x sin 3x } 

_ = 2 < - H -- - . }. 

y (sin y sin 2y sin 3y " ) 

I should doubt its being true. It is a particular case of the theorem 

_^ to n = + oo 



but this is only true when there is no discontinuity in the series 
(j>x + (f> (x + 1 ) .a + Q (x + 2) . a z + . . . . , 

* Svpra, p. 252. 
VOL. III. S 



258 Correspondence between Sir W. Rowan Hamilton and 

which, I should think, would not turn out to be the case in the 
above. 

* I hope you are well, and taking care of yourself. Nobody 
gives you a good character in the second particular. 

* The Astronomer Royal in this country always lays down his 
work the moment he feels wrong, and plays till he feels right 
again. You have too much of our stock of science invested in 
your head to be allowed to commit waste. You are only tenant 
for life, and posterity has the reversion ; and I don't see why you 
should not be compelled to keep yourself in repair.' 

From SIR W. B. HAMILTON to A. DE MORGAN. 

* OBSEEVATOBY, January 5, 1845. 

4 Many thanks for your revise; the final copy will also be 
welcome. 

6 Have you one to spare of your first Paper on the Foundation 
of Algebra ? I know you sent me one, for (too rare event with 
me) I thanked you for it, and remember that I quoted Kant ; but 
it is buried, I am sorry to say, among piles of pamphlets and 
papers, and is, for the present, lost. 

' Is it too late for you to try whether you may like the follow- 
ing geometrical interpretation of your cubic system ? In page 8, 
equations near the foot, is the dihedral angle between two planes, 
one containing the primary unit line (1, 0, 0), and the other the 
line (0, by c\ ^/f I cos 0, */\ I sin 0, and ^/i &*, are three rectan- 
gular co-ordinates. Let a + b = 0, a + c = 0, be called the equations 
of the axis of your system, and a = b + c the equation of your 
equatorial plane. Then the following theorems are true as inter- 
pretations of your rules of multiplication : 

' 1st. The projections of the unit-line, the two factor-lines, and 
the product-line on the axis are proportionals, in the ordinary 
Bense of real or single algebra. 

' 2nd. The projections of the same four lines on the equatorial 
plane are also proportionals, in the sense of Mr. Warren's doullc 
algebra. 

* At least I know that these theorems are true when we take as 
equations of multiplication those other forms which you mention : 



Professor A iigusfus De Morgan. 259 

A = be' + cb f + aa f 
B = ab' + ba' + cc 

C = ac + cd + bl>'. 

What put them into my head I may tell you another time, as I 
hope to write soon again. In great haste,' &c. 

From SIR W. E. HAMILTON to A. DE MORGAN. 

* OBSERVATORY, January 5, 1845. 

1 A medical attendant having called to look at my ankle, which 
is slowly recovering from a sprain, I write a line while he is wrap- 
ping something about it (holding the paper in my hands), to make 
sure of conveying rightly what I sent a while ago to the post in 
Dublin by another messenger, about the meaning of your 0, as I 
wrote then also in great haste. 

' It is the dihedral angle between the planes passing through 
what I call the axis of your system, 

x + y = 0, x + z = 0, 

and containing respectively the lines (1, 0, 0) and (a, b, c). I have 
no copy of what I wrote just now, but hope that this was the 
meaning.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, January 12, 1845. 

* I am much obliged to you for the interpretation in two notes. 
I have been so much occupied with the recommencement of lectures 
that I have not yet had time to consider it properly. I write 
this merely to say that I send the Paper on triple algebra by this 
post, since the receipt of the same without a line might make you 
think your notes had miscarried. 

' I hope your ankle is better.' 

From the SAME to the SAME. 

1 December 23, 1845. 

[After giving a list of Hamilton's Memoirs, in quarto, in his 
possession, Professor De Morgan continues] : 

' . . . I shall be obliged very much if you will tell me at your 
leisure (for the thing does not press at all) whether this is all you 

S2 



260 Correspondence between Sir W. Rowan Hamilton and 

Lave published in quarto. And also whether you have ever 
appeared in octavo elsewhere than in the Notices of the E. I. A. 
and the Keports of the British Association. 

' Should you have a spare copy of anything not above named, 
of course I should be very much obliged to you for it ; but suppos- 
ing your stock exhausted, the name and date will be of use 
I dare say I shall pick up anything, if I know what to look for. 

' If I were you (this by way of side appeal to your own authorial 
feeling) I should always take care to put anybody who is trying to 
be a bibliographe into possession of all correct information; for 
on the " harmless drudges " (as Johnson called lexicographers) must 
depend whether you go down in full or mangled. 

' 1 have had no time for algebra this year not even to read 
Mr. Graves's* Paper thoroughly capital as it is. The " Symbolic 
Geometry" I have only seen, not read yet ; but I am glad to see it. 

' I hope you are in health and with good will to your work. 
You must have a good many shots in the locker.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

* OBSERVATORY, TRINITY COLLEGE, DUBLIN, 
< February 2, 1846. 

' More than a year ago I made a communication to the Eoyal 
Irish Academy, in which your Triplets figured. The notice was 
in type in June last ; for I received and corrected " proofs " in that 
month ; but separate printed copies have never since been received 
by me. The hope of receiving them has been an excuse to my 
conscience for not writing sooner to you ; allow me now to send 
you the manuscript notice, such as the printers returned it to me 
last summer, with all its faults of hasty writing and semi-erasures 
on its head. I shall be much gratified if you accept it, in that 
slate, as some small payment in kind for the proof-sheets of your 
important Paper on Triplets, which you sent me in 1844. You 
Lave probably seen, and I presume have received from the author, 
the notices of the communications made to our Academy here by 
the Eev. Charles Graves on the same general subject last year : 
they pleased me much, and you will find that your priority is 
acknowledged, on several important heads. 



* Professor Charles Graves : see infra. 



Professor Augustus De Morgan . 261 

'For my own part, I have not been exclusively occupied by 
my Quaternions, but confess that they have been growing in 
interest upon me, and that I more and more believe they will one 
day justify a hope which I ventured to express in an Address to 
the E. I. A., on the first night of its session of 1844-5, namely, 
that they will constitute nothing less than " a new algebraical 
geometry." Of that Address, or oral communication, which was 
somewhat extemporaneously made, other authors choosing to yield 
me precedence at the time, I am fortunate enough to possess the 
power of giving some printed notices to scientific friends, having 
ordered the printers to strike off some at my own expense before I 
went to Cambridge last June ; at which University I gave several 
copies away, but have several still remaining, of which I shall be 
happy, if you will allow me, to present you with one. 

* The words above-mentioned do not, I believe, occur in the 
printed notice, but it is a very fair statement (drawn up indeed by 
myself) of the substance of the communication which I made in 
November, 1844. Some later communications have been made by 
me since to the Academy, on the applications of Quaternions to 
Problems of Dynamics and to Surfaces of the Second Degree. 

'If I have alluded to delays in the publication of the Pro- 
ceedings of the Academy, it is only fair to allow that, besides 
my own personal dilatoriness, I may, though not the editor, be in 
some degree to blame for not more strongly urging punctuality as 
President. But I hope that some allowance may be made for my 
unwillingness to have the air of coercing persons even more 
dilatory, and at least as scientific, as myself. 

' I did indeed procure the passing, about a year ago, of a self- 
denying ordinance, by which any author who was not ready with 
his abstract in its turn was to submit to be relegated to an Appen- 
dix, and have acted or suffered thereupon, with reference to some 
of my own communications of last session. The Proceedings of 
that session have been all (with the exception of the said Appendix) 
for some time in print, and I heartily wish that they were published; 
but believe that the editor desires to suppress a certain controversial 
correspondence, which can scarcely interest any beyond the circle 
of our own members, though it could do us no discredit. (It is one 
which in no shape concerns me personally.) 

As to myself, you may perhaps be aware through some extracts 



262 Correspondence between Sir W. Rowan Hamilton and 

from the Dublin papers, or if not, let me be now allowed to inform 
you, that I gave notice at the beginning of the present session 
of an intention (which had been formed after a very careful delibera- 
tion on my part) to retire from the Presidentship at the annual 
election next month (on the 16th of March). I hope that my tenure 
of the chair has not been wholly useless to the body : certainly 
the Academy is now, in numbers and energy, as well as in collec- 
tions, more rich and flourishing than it was when I was elected in 
1837, however little my exertions may have contributed to such a 
result. But I feel it to be quite necessary that I should have more 
leisure for scientific study than the labour, and still more the cares 
of the office, have for several years allowed me; and it is very 
pleasant to me to think that if the Academy shall have any diffi- 
culty respecting my successor, it will only be to make a choice 
among several persons eminently worthy. 

' I have not your last note beside me at this moment, but will 
not let that circumstance serve as an excuse to me for any further 
delay. The accompanying Paper, on the earliest printed Almanacs, 
was very welcome, especially as I had been much interested a 
year ago in the question about the time of Easter, and much 
amused by some of the newspaper correspondence on the subject. 
Indeed I was applied to by an Irish clergyman of some general 
talent, but who had no acquaintance with the subject, and with 
whom I was not myself personally acquainted, to give a decision 
ex cathedra, as Eoyal Astronomer of Ireland. I took some pains 
with my reply, chiefly that I might convey without discourtesy 
my view that he had no right to consult an astronomer on the 
subject at all, but was " concluded " by the tables in the Prayer 
Book (published, I believe, as they now stand, in the Act for 
change of style?) as the authoritative voice of the Church and 
State in the matter. And I remember that among my illustra- 
tions of the inconveniences which would have resulted from the 
absence of such authoritative and technical directions, I took some, 
with acknowledgment, from a letter of yours to the Athcmcum. 
At the same time I pointed attention to a rubric, connected with 
one of the Tables (the one for the 20th century), which had not, 
and has not, been noticed, so far as I know, in the controversy, 
show that a difference between the " ecclesiastical " and the " real " 
full moons was perfectly well known to exist. My worthy 



Professor Augustus De Morgan. 263 

correspondent has never favoured (or troubled) me with an 
answer.* 

4 1 feel some curiosity to know how the Moravians settled the 
point, and shall perhaps consult an excellent old uncle of mine, 
who is a retired Moravian minister, and with whom I correspond 
occasionally on things connected with astronomy (the Rev. John 
Willey of Grracehill). He has a passion for numerical calculation, 
and is just now in raptures with a work of Burckhardt's, of which 
I was lucky enough to ascertain the title for him. 

' Are you aware that the Equinox in 325, A.D., fell on the 20th, 
not on the 21st of March ? Some very simple mental calculations 
led me a few years ago to doubt the latter date, though asserted 
in, I cannot say how many, books on astronomy ; and I then settled 
the point at least to my own satisfaction at the time by a careful 
computation with Vince's and other Solar Tables. I wish you 
would enlighten me as to the history of this widely-spread mistake. 

' P. S. I am happy to say that I have at least one Paper of 
some elaborateness, which, according to my recollection of your last 
note, you do not seem to have ever received, but of which I could 
put even two or three copies at your disposal. It is the " Second 
Part " of my Essay on a General Method in Dynamics. I find that 
it is quite possible, and not difficult, to connect the general formula 
of that Essay with Quaternions. 

' 2nd P. S. I have been dreadfully remiss in writing not only 
to you, but to my old friend John Graves: you may, it is not 
unlikely, meet him before I write ; in that case give my very kind 
remembrance, and tell him, if you please, that I have reserved 
for him a copy of the Abstract which I brought with me to 
Cambridge.' 

From A. DE MORGAN to SIR WILLIAM HAMILTON. 

' 7, CAMDEN-STREET, CAMDKN TOWN, 
< February 15, 1846. 

'Your note was not to be answered off-hand, and when I 
received it I was as busy as a person need be in concocting an 
Annual Eeport for the Astronomical Society. 

' But now it's read, and that's all over, and I shall neither lay 

* See vol. ii., pp. 482-4. 



264 Correspondence between Sir W. Rowan Hamilton and 

down and cry nor die in consequence. In fact I made a memoran- 
dum to survive to answer yours. 

* First, I thank you for the notice both as to matter and manner. 
The former I shall con in detail the first time I can get again on 
triplets ; and the MSS. will bind up with the other things very 
nicely, and increase the value of the volume. 

' I was very much gratified with C. Graves's Paper, but, as 
before, have never found the opportunity to try it as I could wish. 
But it is a finished thing. I rather suspect there is another ; but 
that one is that one, and must continue to be so. 

'The Quaternions will, I have no doubt, make a system, in 
which rotations round the three axes play the part of co-ordinates* 
complete in itself. I shall be very much obliged to you for the 
notice you mention that which you distributed at Cambridge. I 
heard speak of it, but did not see it. 

4 1 was glad to hear that you are going to resign the President- 
ship. You have no business there at all there are plenty of 
people who can do all that a President, as such, has to do ; and I 
maintain that any man who is fit for original research has no 
business to be a president or secretary or treasurer, at the expense of 
his researches. 

' Now for Easter. I am not sure, by your note, that you saw 
my first communication to the Companion to the Almanack on this 
subject, containing the development of what you mention in the 
AthencBum. I therefore send one. Should I have sent it before, 
you can find some one to give it to your uncle, for instance, when 
you ask him for the Moravian determination. 

',You got him Burckhardt ; no doubt B.'s table of prime num- 
bers up to 3 millions odd. But B. only gives the fact of primeness, 
or, when not prime, the lowest divisor next above 1. But Chernao 
Cribrum Arithmeticum, Daventriae (Deventer), 1811, 4to, gives, up to 
1,020,000, every prime factor, which is more convenient. 

' As to the equinox of 325, it is a not very well known fact that 
using the Alphonsine Tables to recover Easter of the Nicene 
Council, the Gregorian Committee made a mistake of a day, and 
concluded that the equinox ought to vibrate over the 21st and 
22nd, when in truth it did vibrate from the 20th to the 21st. 
Their equinox of 325 was a theoretical one formed from their own 
Tables. The mistake hangs on to what I have been pointing out 



Professor Augustus De Morgan. 265 

in what I send and sent, namely, that a specific astronomical code 
has been forged for the Nicene Council, though not one word was 
said by them except "Easterns, keep Easter as the Westerns do." 
But what the Westerns did whether it was moon, sun, or neither, 
which regulated; whether even the Council knew the difference 
between the Sun and Moon cannot be gathered from what they 
said of Easter, or what any contemporary historian has said. No 
wonder that the equinoctial 21st should have been fastened on 
them, as well as the cycle of Dionysius, or anything else which was 
made after them. 

1 1 have not seen Graves lately, but hope to do it soon. I shall 
be much obliged to you for your second part of Dynamics. I now 
remember to have seen it ; but when I wrote to you I mentioned 
nothing but what was before me. But none of your "some 
elaborateness." I want a complete list ; and when my note turns 
up again pray answer me categorically, as they say ; but there is 
no hurry. In the meanwhile, if you will just send the second part of 
Dynamics and the Cambridge Abstract, and Grraves's copy of it, 
which you mention, in a railroad parcel, addressed to me at " Uni- 
versity College, London," I shall be very much obliged. 

' 1 have been trying lately at Arbogast's method of derivations, 
and find that there is much to learn out of him. 

* The hindrance in the way of development (I mean of organized 
rules for it) appears to have lain in the imperfect view taken of 
Taylor's Theorem. 

* If we write 

(a + b) = $a + fy'a . b + $"ci +..., 

<e 

the 2, 2-3, &c., stick in the way of describing these terms as 
formed upon a repetition of one process. But if we write it as 

6 



d f 


da J 



__ 77 

db d>a + db 

da 



we see that there is one operation which, being successively per- 
formed on 00, will produce the successive terms. 

' Derivation, the real tool of development, is not a mere differen- 
tiation, as Arbogast would at first have made it, but a differentiation 
accompanied by a subsequent integration ; and in the series 

a + bx + CvC 2 + ex* + . 



266 Correspondence between Sir W. Rowan Hamilton and 
the tools to be used in developing functions of it are 



the successive uses of these giving what Arbogast calls his dtriv&s 
divisees ought to have been his dtrMes. 

' I have sent this view of the matter, with a new mode of de- 
monstration, to the Cambridge Mathematical Journal, where, by the 
way, I saw your Algebraical Geometry. A little more attention 
to this subject would perhaps extend the calculus of operations. 
Arbogast's divided derivation (call it D) gives 

(a + Da . x 
D<f>a . x 

in which a, Da, D*a, are independent. This is equivalent to saying 
that if v be the operation 



and <J> any functional symbol ; then 

(V) = V 
But it is not true that 



for reasons connected with the effect of v in introducing new and 
independent subjects for the second v to deal with. 
' But it grows late, and I must therefore conclude.' 

From the SAME to the SAME. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
' April 12, 1847. 

* I send you a Paper on Logic, out of which will arise a question 
of literary piracy. I wish you would give me your opinion on this 
point (I, you understand, am the asserted pirate). 

* Looking at 3, on the <jt(n(/ft/ of propositions (admitted to be 
mine), what hint did I need to write thereupon the first two pages 
of the addition at the end ? 

1 The party making the charge is your namesake, Sir William 
Hamilton of Edinburgh. If I cannot drive him to press in a v 



Professor A ugustns De Morga n. 267 

or two (which I am trying at, but he does not answer the spur as 
well as a man ought to do who has made such a charge but then 
his health is not good), I must publish myself ; so that all will 
soon be out. 

* In the meanwhile I send you this information that you may 
not stare if anybody tells you that you are charging me with steal- 
ing logic from you. I shall take every possible care to identify 
my man and distinguish him from you ; but I know it will not 
entirely succeed, for you are constantly confounded with the 
Edinburgh Sir W. H. 

' 1 was talking to a friend on this matter the other day, and I 
said to him, " You know Sir W. H. is no mathematician in fact 
he is an opponent of mathematics." I saw my friend's eyes open 
very wide, and he looked to see if I were gone mad I had for- 
gotten to say " of Edinburgh." : 

From SIR W. K. HAMILTON to A. DE MORGAN. 
[FROM A COPY CORRECTED BY SIR w. R. H.] 

1 OBSEBVATORY, DUBLIN, May 7, 1847. 

* I have received your two recent communications, and think 
you have taken all reasonable precautions against my being con- 
founded on the present occasion with my celebrated namesake of 
Edinburgh. I daresay it would be nuts to you to have two Sir 
William Hamiltons on your hands as controversialists at one time ; 
but this note is to warn you that I don't think I shall indulge you 
on that point. Perhaps you may ask, What provocation have you 
given to such a simultaneous controversy ? A very gentle one, cer- 
tainly, and not very recent, but one which I might have a good 
opportunity of now accepting, if I were disposed, which I am not. 
My manuscript researches respecting Quaternions, and their applica- 
tions to Greometry and Physics, having attained a considerable extent, 
and a number of scattered notices, themselves by this time not very 
small in bulk, having been printed, I am urged by my friends in 
Dublin, and am myself now desirous, to make at least a bey inning 
of that more full and formal publication which I have all along 
intended. After many hesitations as to whether I should not at 
once proceed to the parts which are more likely to interest and 
not to shock mathematical readers in general, I have decided on 



268 Correspondence between Sir W. Rowan Hamilton and 

following a more historical order; and have handed in to the 
Committee of Publication of the Royal Irish Academy, who have 
transmitted to their printers a Paper entitled Researches respecting 
Quaternions, First Series; in which Paper I have endeavoured to 
insert nothing with the principles of which I was not familiar at the 
time of making my first communication to the Academy on Qua- 
ternions, on the 13th of November, 1843. A good part of this 
Paper has been lying by me for some considerable time, and on 
the whole it is, I think, very nearly what I would have drawn up 
in 1843, if our rules had been stringent enough to oblige me as an 
author then to hand in at the time a MS. prepared for being put 
into the printer's hands. The word " Triplet " does not occur once 
in this Paper, but the word "Set" presents itself very frequently ; 
because I had in fact been familiar with the conception of sets, as 
including the conception of couples, for at least nine years previous 
to my perceiving my own definite system of Quaternions in October, 
1843, and had announced an intention of publishing hereafter a 
theory of triplets and sets of moments, steps, and numbers, which 
should include the theory of couples, when I published (about 
August, 1835) in the xvnth. vol. of the Transactions of the Eoyal 
Irish Academy, my Paper on Algebraic Couples, and on Algebra as 
the Science of Pure Time. See the concluding sentence of my Essay 
on that subject in that volume. 

* Now for the controversy which I think is not to take place. 
It does not relate to the Triplets, on which my old unpublished 
and rejected researches cannot and ought not to interfere with your 
priority. But you may remember or may not, for an author so 
original and fertile as yourself has room for forgetting many things 
of his own which other people find it worth while to remember 
that in the first of your Cambridge Papers On the Foundation of 
Algebra you expressed, though very politely, a certain degree of 
dissent from my general philosophical (or if you choose w/philoso- 
phical) view of the subject. I am conscious of having expro 
Hint view, such as it was, obscurely at the time, nor have I much 
hope of being able to express it more clearly now, without taking 
more trouble and occupying more room than can perhaps be wrll 
spared from other things at present. But as I have not yet re- 
jected it, and as the Quaternions did really arise in my own mi ml 
one day that, being then fresh from a reperusal of my old Essay, I 



Professor Augustus De Morgan. 269 

renewed my attempts to combine my general notion of Sets of 
Numbers, considered as suggested by Sets of Moments of Time, 
with geometrical considerations of points and lines in tridimensional 
space, it has appeared to me to be the most natural, clear, and 
honest course, to print, as the First Series of my Researches on 
Quaternions, an account of the manner in which the mathematical 
notion of Time leads (in my mind at least) to a general conception 
of Numerical Sets, which has by me, as yet, been only exemplified, 
in anything like a satisfactory and definite way, for the two cases 
of Couplets and Quaternions. Eespecting the geometrical applica- 
tions, the printers have in their hands for that First Series, less 
than I did actually communicate in November, 1843, by speech, 
and by large diagrams which were then exhibited to the Academy, 
because I am reserving most of the geometry for the Second Series, 
communicated in November, 1844 ; to be followed by a Third 
Series, of a more dynamical character, of which sketches were 
given to the Academy in 1845 ; and these, too, probably by others 
with the list of which I forbear from now alarming your patience. 
And although it is likely that I shall append to the First Series 
some general remarks, as yet unwritten, and to be dated, as an 
appendix, according to the actual time of writing them, which will 
probably not be until the printers are actually ready ; yet I do not 
think it likely at present that I shall write anything of a contro- 
versial character in such appended and general remarks, which 
abstinence from controversy, if it be realised, will not I hope be 
accounted disrespectful by you. Metaphysical and logical specu- 
lations have a great charm for me, but in a certain sense and degree 
my mind is (I think) less analytical than synthetical ; that is to say, 
as bearing on the present subject ; whenever I catch, or fancy that 
I catch, a glimpse of a principle, I am impatient to apply it not 
exactly towards the making of a railroad, but still to apply it in 
some way of my own. Thus, I like better to work out my notion 
of Time into its mathematical consequences, than to enter into any 
d priori discussion whether it be metaphysically correct, though I 
have speculated on that point too. And without pretending to 
settle by any clear definition beforehand what symbolical geometry 
should be, I have been gradually working into shape, by trial upon 
mathematical questions, my idea of symbolical geometry. 

' Unlike as my little Papers on this latter subject in the Cam- 



270 Correspondence between Sir W. Rowan Hamilton and 

bridge and Dublin Mathematical Journal may appear to those other 
Papers which I have hitherto printed on Quaternions, yet, if you 
have dipped into both, you will not long fail to recognise, perhaps 
may have recognised already, that the " Geometrical Fraction " of 
the one set is just the " Quaternion " of the other in disguise. 

'I go, you know, very seldom to London, but was there for 
a while last August, or at least at Clapham, where I was leaving 
my eldest boy at Mr. Pritchard's school. Your residence was 
rather far, but I several times intended to call, though not sure of 
finding you at home ; and the last thing I did before returning to 
Ireland was to make up a small packet of a few printed Papers, 
which I had designed to hand you, and which (I hope) reached 
you by post soon afterwards. I am far from being entitled to 
expect others to acknowledge the arrival of such things from me, 
since I seldom expressly and at the time acknowledge the many 
valuable Papers which I receive from other authors; but if you 
are writing to me on any other subject, I should be glad to know 
whether you received, about Christmas last, a larger octavo 
pamphlet on Quaternions, and a little Paper in January, On 
the Law of the Circular Hodograph to which I have since been 
able to add several general theorems the Quaternions which have 
long since become a calculus (= hobby ?) in my hands assisting me 
most materially in the investigations. My reason for asking is 
that I would try to forward other copies, if those former ones did 
not arrive. I remain, &c. 

' P. S. I make up a printed copy (first sheet of No. 50 of 
Proceedings) of some of Charles Graves' s remarks on Triplets, and 
of my Interpretation, &c. 

* Received a long and interesting letter from you about March 
or April, last year. Unlucky enough not to be able to find your 
Paper on Triplets perhaps I may like to refer to it in my 
"Appendix" to the "First Series" at all events am reading 
up in several directions, and should be glad even of the loan of the 
Paper. However, I am much interested at present in some appli- 
cations of Quaternions to Physical Astronomy.' 



Professor A HO us! us De Morgan. 27 r 

From the SAME to the SAME. 

'OBSERVATORY, July 10, 1847. 

* On Thursday evening, July 1st, about 7 or 8 o'clock, at the 
Euston Hotel, London, near the railway station, before starting 
for Liverpool (or rather for Birkenhead), on my return homeward 
from the Oxford Meeting, I directed to your residence (7, Camden- 
street, Camden Town) a quarto copy of my First Series of Researches 
respecting Quaternions, and paid sixpence for its safe delivery; I 
therefore hope that it has reached you. If I had remained in 
London even one clear day, instead of remaining only an hour or 
two, I would have waited on you with the Paper. 

* Am I wrong in thinking, or feeling, it to be somewhat 
remarkable that a false alarm respecting a relative of mine, 
arising from a mistake of names, should have hurried me home 
from the Oxford Meeting in 1832 ; and that a true account of 
the death of the same aged relative, whose funeral in Ireland I 
was in time to attend last Saturday, should have again caused me 
to hasten home from another Oxford Meeting in 1847 ; each time, 
it is true, just about the close of the Meeting ? the relative in 
question having been generally a remarkably healthy man, and no 
uneasiness felt about his health until extremely recently, except 
when a false report reached me at Oxford, at the time above 
alluded to, that he had been attacked by cholera fifteen years ago. 
It shows perhaps a predisposition to superstition that the coinci- 
dence should strike me as it does. 

' I have the strongest hopes that the Quaternion Calculus will 
simplify physical astronomy, especially in all those departments 
in which spherical trigonometry is now employed. But as yet 
I don't pretend to have done more than to have proved that my 
Calculus does really take hold of the subject : that it is adequate 
to work out new and true expressions for the perturbations of the 
moon and planets which agree with known results, when translated 
into the known forms of language. Whether a sufficient compen- 
sation is afforded for the trouble of acquiring some new habits of 
calculation, and giving up some old ones, I am not an impartial, 
and therefore not a competent judge. But it is my business to 
multiply the materials for others to form their judgment upon; 
and this, if it be an arduous, is also a delightful task. The 



272 Correspondence between Sir W. Rowan Hamilton and 

freshness which old subjects receive, at least to me, from my new 
view of them, is a charm sufficient to repay me for any amount of 
labour.' 

From A. DE MORGAN to SIR "W. E. HAMILTON. 



, 1847. 

' I have received the Paper, for which thanks. When I am a 
little out of my logical undertaking I hope to read it attentively. 

' I hope you remember that you have to send me the second 
part, I think, of your dynamical Paper, which, apropos of the list 
of your works which I sent, you found I had not got. 

4 Your coincidence is a curious one that your relative should 
never be either truly or falsely said to be dangerously ill, except 
to hurry you from the Meetings at Oxford alone. There can of 
course be no discoverable connexion between the two things. That 
there is no connexion is more than I know or you either. 

' I am myself past thinking anything too extraordinary to be 
true. If we knew everything, should we or should we not find 
that all things are connected? that every action of every man 
that ever lived is connected with every action of every other man 
that ever lived ? 

' We know that every motion of every particle of matter has 
its effect upon the motion of every other particle. 

4 If the law of attraction be veritably and physically true, such 
must be the case. One man was so staggered by the idea that his 
snuff attracted the snuff in the Saturnian snuffboxes, that he wrote 
a book against gravitation. 

' Now is there a mental dynamics ? I can't tell. In these 
matters I can admit the possibility of anything, as long as a 
man says he can't prove it. 

' There are two things which want a good deal of consideration 

The action of moonlight, 
The stories of ghosts. 

I never reject the whole world's opinion entirely. In all nations 
it is a maxim that there are certain things not to be done in 
gardening when the moon is growing; and in all nations thnv 
are independent assertions of phenomena marking the deaths of 
friends to the absent. The two things stand alike ; there is for 



Professor Augustus De Morgan . 273 

"both independent and universal tradition, constantly affirmed to 
be reinforced by fresh observations ; for both, vulgar exaggeration ; 
for both, the almost universal rejection of philosophers; for both, 
continual exceptions to the rule of rejection on the part of educated 
men ; for both, cases of isolated facts, in the evidence for which 
the flaws seem to be invented, not discovered. And so I leave them 
both. 

* I have almost forgotten all about triple algebra. I have no 
doubt of the applicability of the quaternions. How can a compli- 
cated and self-consistent system fail to represent complicated things 
with relative ease ? ' 



From the SAME to the SAME. 

'Auyust 17, 1848. 

'I am glad to hear of your generally better health report, 
backed by Graves, asserts it. 

'Now to my subject. Long time ago I wrote you a list of 
your own works in my possession and asked if it was complete, to 
which you replied, " No, the second part of the Method of Dy- 
namics is wanted, which I will send you." Now, if you made a 
Jesuitical reservation in your own mind and added to yourself 
When the Union is repealed all I can say is, that such reference 
in Kalendas Hibcrnicas leads to odious comparisons; and I think 
to myself, Why the other Sir William Hamilton would have used 
me better than that. 

' Here have your tracts been lying in a heap, of which I am 
now sending twenty volumes to the binder, when they might have 
been vertically shelved in all the honour of leather and the glory 
of gilding. 

'Pray think of my case, or rather of the impossibility of 
putting a case on the matter of this letter. And if you cannot 
find a copy, send me at least a reference to the volume of the 
Philosophical Transactions. 

* For as to binding an imperfect set a bibliographical con- 
science revolts at it, and an Irishman in these days ought to 
sympathize with revolt of any kind. Or if you are ever so strong 
a loyalist, you ought to allow your tracts to feel a sense of obliga- 

VOL. III. T 



274 Correspondence between Sir W. Rowan Hamilton and 

tion as well as yourself, which they cannot do as long as they are 
unbound. 

'I suppose this letter will find you on your return from 
Swansea, if you are there.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

OBSERVATORY, August 21, 1848. 

* I am really flattered by your wishing to have that copy of 
my Second Essay on a General Method in Dynamics. I own I 
think it has some value, but one thing puts another out of my 
head. After some years, or possibly even sooner, if my health, 
which is quite strong at present, lasts, I may be induced to resume 
the subject, with such lights as later reflection may have given, 
but I feel that it will be a shame if I do not publish, after some 
time, a quaternionic treatise on mechanics ; for / account quater- 
nions an instrument, and not a toy. 

'Wonderful to relate, however, the quaternions have been 
almost entirely out of my head for the last month or six weeks, 
but I am going to attack them again, and indeed Charles Graves 
has been pressing me to produce some book, large or small, on the 
subject, which he can desire mathematical students in Dublin to 
buy. The little interruption to my thoughts, which has arisen 
partly from some private causes (including the illness of a relation,, 
and partly from some anxiety about the state of this country, may 
turn out to have been of service to me/ 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 April 28, 1849. 

' Mr. Erck brought me your Papers, for which thanks. I ha\v 
asked him to come to the Astronomical Society, where he will find 
an atmosphere full of notions which will do him good. He s< 
to have an idea that a man with a long head and a short equatorial 
can do nothing to the purpose : he must be taught better. We 
have men here, or have had, who would have been all the better of 
ehortening their telescopes, if the piece cut off could havo btvn 
added to their working hours. What do you say, as a meta- 
physician, to a piece of length cut off and added to time P 






Professor Augustus De Morgan. 275 

'Tell the boys that I am quite of opinion that co- co- is 1, or 
rather that a co- co- is (co-) 2 . But I should like them to ex- 
plain this : How is it that co- co" drinks as well at a round tabl> 
as at a square one ? And how is it that the cuckoo is not a squ 
bird, but remarkably squat ? Is it the little variation of spelling 
that does all this, or has it rather reference to change of moani- 
This is a puzzle for them. I had no idea you had sons old enough 
to be trigonometrical. 

' I have often used the phrase co-cosine in my lectures. 

' I am printing nothing at this moment, save a solution of the 
old trial equation 

x 3 - 2x = 5 

done by a pupil of mine, and verified by another euqation, to 103 
places. To wit, 

x = 2-09455148154232659148238654057930 
29638573061056282391803041285290 
45312189983483667146267281777157 
757859 +. 

Another tried 150 places, but broke down at the 76th, which was 
wrong. 

1 There is a great deal to be done in organization of processes 
but not after midnight so I subscribe myself, the right to date as 
below* having just accrued.' 

From the SAME to the SAME. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
< August 30, 1849. 

* You are perhaps aware that, in the new arrangements for the 
Colleges, Young has lost his situation at Belfast ; but you may 
not perhaps be aware that it has utterly ruined him, and that 
he and his family will probably be without any means of support. 

* I hope you know and think enough of his long exertions to 
be willing to use what you have of influence which ought to be 
something in such a matter to induce Government to do some- 
thing for him, cut out as he is by their -means from his subsistence. 
I have written to Mr. Tennent, the Member for Belfast, whom I 

* Professor De Morgan dates his letters at the end. 
T 2 



276 Correspondence between Sir W. Rowan Hamilton and 

kne/w long ago, and should he feel able to make any application I 
hope you will feel able to back it. It seems to me a very cruel 
case, and one which might easily be avoided, for I should suppose 
there are many things within Government patronage for which 
Young is quite fit. 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, September 24, 1849. 

'I hope you received what was, almost literally, only a line 
from me, in answer to, or at least in acknowledgment of, your 
letter respecting Professor Young. Although, having never been 
consulted as to any of the details of the new Colleges, I have 
not made myself so far acquainted with them as, even yet, ta 
understand how Professor Young has lost any situation by the new 
arrangements ; yet I am too well informed, through Mr. Spiller 
and through you, of the fact of his being deeply injured, to have 
any doubt on the matter. 

4 At the very moment when your letter arrived I was deep in 
conversation with Mr. Hargreave on the subject ; and had just 
been asking him to join me and others, in backing any memorial 
to Government which may be drawn up by any competent hand in 
favour of Professor Young ; which he expressed himself as willing 
to do. I mean the mathematician, medallist, and now Commis- 
sioner Hargreave, who lately made me a visit. His notion was 
that the new College in Belfast could only injure Professor Young 
by tending to introduce a formidable rivalry; but there must, 
from what I hear, be something more injurious to him than this, 
though I cannot yet understand ichat it is. I shall be most willing 
to join you and others in backing his claims on the Government 
with whom, however, I have no pretensions to exert influence^ 
although I have always met with courtesy at their hands. 

* I have to thank you for a former note respecting Mr. Eivk, 
who was much obliged by your attention, and expressed himself 
to me as having not only enjoyed but profited by his visit to 
London. 

'I remember that in a hasty line, forwarded by him, I s:iil 
something to you about " co-," as a symbolic square root of unity. 
More lately, I was assisting my eldest boy, who is more than 



Professor A u gust us De Morga n. 277 

fifteen now, to form the equation of a perpendicular by co-ordi- 
nates; and he saw clearly that if two perpendicular right lines 
through the origin have for equations 

y = tx, y = tx, 

and if we write t f =ft, we shall have also t =ft' y and .*. f*t = t, 
f- = 1. But here he was disposed to think that we must have 
/ = 1 ; whereupon I took the occasion to give him some notion 
of the Calculus of Functions, and mentioned your work thereon. 
Dr. Peacock remarked to me four years ago that Functions and 
Quaternions have several analogies ... I have, you know, infinitely 
iiKtni/ solutions of the equation p* = - 1 ; namely, those included in 
the form p = ix +jy + kz, where xyz need only satisfy the one 
condition 

x* + tf + s 2 = \: 

From A. DE MORGAN to SIR W. R. HAMILTON. 

' 7, CAMDEN-STREKT, CAMDEN TOWN, 
4 September 26, 1849. 

' The manner in which poor Young is ruined is as follows : 
The Belfast Institution, in which he was Professor, was supported 
partly by Grovernment allowances, partly by pupils. On the 
foundation of the new Grovernment College, the Grovernment 
allowances are withdrawn, and the managers dissolve the Institu- 
tion. So that his situation is in fact abolished and directly by 
the act of Government. 

* Any one would have supposed that a man of his name and 
worth would have been allowed to step from one place into the 
other. But poor Young is no partizan ; and in the focus of 
religious dissension in which he lives such a person has no 
friends. 

* Hargreave is my old pupil, and is in every point of view an 
acquisition to the acquaintance of anyone. 

'What has not analogy with functional calculus? In fact 
algebra itself is a functional calculus. The tf's and b'& are, if you 
like, symbols of operation, not of quantity, the subject of operation 
being the concealed unit. 

' I send you a few loose thoughts : 

' 1. If / (x, y> s) be homogeneous with respect to the three 



278 Correspondence between Sir W. Rowan Hamilton and 

letters, we know that 

/ (x + a, y + b, z + c) = 

is a cone. But we know that if 0, b, c be infinite, this cone is a 
cylinder, and has for equation 

/ (px + qy + rz, p'x + q'y + rz) = 0. 

How are we to show by pure algebra that the first form takes the 
second, when a, b, c are infinite, or any one of them ? ' 

[The remaining " loose thoughts " are concerning 

2. The number of solutions of the indeterminate equation 
ax + by = c, and the limit for c = ab - a - b beyond which the 
equation always admits of solution. 

3. The meaning of the total area of a closed circuit, in the 
case of a curve which intersects itself any number of times.* 

' 1 find the result which follows : 

* Take a point A (outside the whole circuit is most convenient, 
but the modification of the rule for an inside point is easy) outside 
the circuit. Take the element (infinitely small) of which you want 
to settle hoiv often it goes positively into the area, and how often 
negatively, and draw a line through that element and the outside 
point. 

* Name a positive and negative direction of revolution : go 
round the circuit, and every time you cross the test-line mark the 
direction in which you are revolving round A. Take the balance 
of + and - on one side of the element to be tested (either side will 
do). Suppose the balance is 3. Then the element, say dV either 
enters as + 3</For - 3o?F. Take every element, either choosing 
your balance always on the off-side from A, or always on the near 
side, and integration gives the area ; + in one case, - in the other. 

* The theorem is, that though your circuit is as long and as 
tortuous (you make parentheses, and so can I : I never began but 
one chancery suit, and that in the court of worst reputation of all, 
the Irish; and the moment the bill was filed, my party prodiunl 
his accounts, and altered every unsatisfactory item on demand) as 

* In the C<i/it/<ri</</e and Dublin Mathematical Journal, 1850, Vol. v. pp. 
139-142, there ia a Paper by A. De Morgan on this subject, dated Octal 
1849, as U indicated in his letter, dated October 11, is in. 



Professor Augustus De Morga u. 279 



a Chancery suit, the balance is always the same number for the 
same element.* 

'4. This area question would require what may be called a 
swing-swang integration 



where J signifies integration forwards from a to j3, back from ]3 to 7, 
forwards from 7 to S, and so on.' 

This applies to the question of the greatest area on a given 
chord bounded by a given length of arc, in the cases that the circu- 
lar arc which is found is greater or not than a semicircle.] 

From SIR W. E. HAMILTON to A. DE MORGAN. 

1 OBSERVATORT, October 3, 1849. 

4 If I lay a letter out of my hands for a few hours, without 
answering it, I am sure to find that it has been swept away and 
covered up, for the time, by the Charybdis of my other papers. 
No doubt, every such missing treasure may be expected, at some 
future time, to emerge to view ; and may then be suddenly seized, 
by a bold and ready hand. Thus, from month to month, or at 
least from year to year, I find a note or two of yours eddying 
upward to the light; but, for the instant, your last long (and 
welcome) letter is invisible. However, I remember much of its 
contents, and shall send something now in answer to one, at least 
of its " loose thoughts." 

' If we have a homogeneous equation 



we may put it under the form 

/' (P* + <!!/ + rz > P x + 0V + *'*> P"V + tf'y + r"z) = 0, 
where pqrpqrp"q"r" are arbitrary but constant co-efficients, and 
/' is still a homogeneous function of three variables ; or we may 
write 

fix v ~] - F( P* + W + r 
' ' 



p"x + q"y + r"z' 



where the function F is not obliged to be homogeneous. And 

* Note that in a real Chancery suit the balance generally varies with the 
point of view from which the case is looked at. 



280 Correspondence between Sir W. Rowan Hamilton and 
nothing hinders us to assume 



when t is a new arbitrary multiplier, and is a new arbitrary 
function of tico independent variables. Thus we shall have the 
transformation, 

px + qy + rz p'x + qy + r'z 



when / is still homogeneous relatively to its three variables, but 
^ is an arbitrary function, and p . . . r" and t are ten arbitrary and 
independent constants. Change now (as you desire) #, y, s, to x + a, 
// + b, z + c, where 0, b, c are three new arbitrary constants ; and 
suppose these to be connected with the ten preceding constants 
by the three relations, 

pa + qb + re = 0, p'a + q'b + re = 0, p"a + q"b + r"c = t, 
we shall thus have the new equation of transformation, 
/ (x + 0, y + b, z + c) 

(px + qy + rz p'x + q'y + r'z \ 

1 + t- 1 (p'x + q"y + r'z) ' 1 + t 1 (p'x + q"y + r'z))' 

Conceive next that 0, b, c, t, together tend to infinity, still satisfying 
the three conditions lately written, while x, y^ s, and pqrp'q'r'p"q"r' 
continue finite ; the limiting result will be the following : 

/ (x + 0, y + b y z + c) = (px + qy + rz, p'x + qy + r'z) 9 

which seems to meet your requisition, since it does not expressly 
refer to geometry, though you will not fail to see that the process 
was suggested to me by geometrical considerations, namely, by the 
conception of a cone reduced to a cylinder by being thinned while 
its vertex is remc 

1 If we neglected so to thin the cone, we should only mtymfy it 
into a vast helmet in some Castle of Otranto ; whereas we want to 
convert it into a gigantic extinguisher, fit to be clapped upon a 
wick unsnuffed for an eternity, and by consequence infinitely /on;/. 
If you laugh at the bull, don't forget that it would be just as 



Professor Augustus DC Morgan. 2 8 1 

much a blunder if any beginner were to fancy that (for example) 
the equilateral or rectangular cone of revolution, 

x* + y* - z* = 0, 

<5ould ever become a cylinder, by the mere remotion of its vertex. 
The surface 

a? + if - (z + c) 2 = 

is still an equilateral cone, however large the constant c may be ; 
but if we thin this cone indefinitely, by (for instance) introducing 
the factor 6~ 2 into its last term, and thus writing 

a? + y*- c z (z + c) 2 = 0, 

then, indeed, the limiting result will be a cylinder, namely, 

& + if - 1 = 0. 

* Lest this letter should either grow like the wick of my country- 
man's candle, or on the other hand be quite blown out, and fail to 
reach your eyes at all, by any incipient procrastination, I shall 
send it away at once. 

'I have not been in Dublin since, to forward my batch of 
Abstracts. 

' A Dublin paper, which some time ago accused Mr. Hargreave 
of being young, has lately been saying that Baron Richards is 
here, Doctor Longfield there, and Mr. Hargreave nowhere* Do you 
happen to know his present address ? I was very glad to meet 
him.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
< October 5, 1849. 

* Your solution is exactly the thing some commodious way of 
keeping the lower end of the cone within bounds was just what I 
wanted. 

* I have heard from the Member for Belfast, who will support 
the memorial (and has supported the claim already) in favour of 
the disbanded professors. He thinks it will be referred by Lord 
John Eussell to Lord Clarendon. 

' I do not know Hargreave's address. 

* Commissioners of the Encumbered Estates Court. 



282 Correspondence between Sir W. Rowan Hamilton and 

[He then returns to the question of the area of a circuit as it 
is discussed in the Paper in the Cambridge and Dublin Journal 
of Mathematics, already cited, and concludes] : 

* For your boys you may put it thus : 

* To puzzle a churchwarden, let the black line be the boundary 
of the parish, then the parts shaded red are out of the parish. 

' If a part marked 1 pay tithe, then (2) pays double tithe ; but 
to the inhabitants of - 1 the parson pays tithe. 

* If a boy driving his hoop in 1 is to have one cut from the 
beadle's cane, then a boy doing the same in (2) is to have two cuts ; 
but a boy so employed in - 1 may give the beadle a cut, I think 
with his own cane, what does the formula say ? May the boy use 
his own hoopstick ? ' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

OBSERVATOEY OF T. C. D., October 10, 1849. 

* Many thanks for your figure, which I am sure will be worth 
studying. A little daughter of mine, about nine years old, who 
has a formula for anything new she learns from me, " Deeply 
interesting, combined with being deeply curious and instructive," 
after gazing for some time on your "walls of Troy" so my 
schoolboy companions used to call a labyrinth on a slate or paper 
commenced her usual exclamation, and got as far as " Deeply 
interesting, combined with being deeply curious " ; but there 
stopped short, and said, " I cannot add, and instructive, for I do 
not understand it at all." 

* Her papa is in a little better condition on the subject, and that 
is all he can say, as yet. But in earnest, from such attention as I 
have hitherto given to the matter, I think that the speculation is 
not unlikely to have even practical fruits; nor would it at all 
surprise me if some electrical phenomena of circuits were found to- 
correspond to your law. 

'The remarks on integration reminded me of a mode of proving 
by quaternions an important but known proposition respecting 
polyhedra, which corresponds to the physical possibility (or fact) of 
equilibrium of a closed body immersed in a fluid of the same 
specific gravity ; but though you were one of the first persons who 
noticed the Quaternions, the proof might merely bore you. 



Professor Augustus De Morgan. 283 



' The enunciations on the first half of this sheet may, however. 
interest you, though the Athenceum leaves them unreported as 
abstruse* I own I should be curious to see (what no doubt can be 
given) a demonstration of either theorem by co-ordinates. 

* You saw, of course, what (oddly enough) I did not at once 
perceive myself, that I was tacitly assuming your homogeneous 
function / to be of dimension zero ; but this seems not to limit the 
essential generality of the question. 

* I have not yet got to Dublin to send off my parcel for you. 
The enclosed note will show that I was not allowed to indulge myself 
respecting Professor Young in the way I proposed ; but I may still 
be of some little use, by signing a memorial/ 

From A. DE MORGAN to SIR W. K. HAMILTON. 

' 7 CAMDEN-STREET, CAMDEN TOWN, 
October 11, 1849. 

* Nothing about Quaternions will bore me, if I can only make 
it bore through me. Ink must be cheap in Ireland if you can 
afford to waste it on such a supposition as that. 

*I have thought more than once about electric currents in 
reference to my area conundrum. I suppose the qualitative 
character of formulse expressing the action of a current on a point 
must depend upon the law of signs I have given in some way. 

* But I am a searcher after things mental not material and 
to me, whose business of life it is to study the development of the 
mind the interpretation of 



T TO y,, l+1 - y m 

is practical while the progress of electricity through a wire is 
comparatively theoretical. 

' The demonstration is very easy. I hope to have it in print 
in the next number of the Cambridge and Dublin Mathematical 
Journal. 

' I could make a teetotum game out of it for your little girl. 

' I agree with the Athemeuw, that your theorems are abstruse ; 
but they should have reported them for those who like abstruse 



* Respecting inscription of gauche polygons in surfaces of the second order. 
Printed in Proceedings of the R. I. A., for June 25, 1849. 



284 Correspondence between Sir W. Rowan Hamilton and 



things. It is a case in which geometry must teach algebra a 
demonstration. 

' Your theorem ought not to subdivide. The odd polygon 
ought to be the even one, with one evanescent side. Pascal's 
theorem about the hexagon is equally a theorem for a 5- or 4- or 3- 
sided figure. For instance, draw a A and tangents at the angular 
points. These tangents are the directions of evanescent sides of 
a hexagon, and the theorem of the intersections remains. 

' I have not heard anything more of poor Young/ 

From the SAME to the SAME. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 November 12, 1849. 

' Your parcel received this day by Mr. Erck ; and I await your 
instructions as to the disposal of the copies for which you have a 
disposal. The rest I will take care shall go to people who are up 
to such things. 

' Here is an enigma for your little girl : Who ought to have his 
heart in his head ? Answer A teacher ; for, primo, he ought to 
have his heart in his subject ; secundo, he ought to have his sub- 
ject in his head ; ergo, he ought to have his heart in his head. 
The only objection I have heard is from a teacher of anatomy, 
who objects very much to having any heart in his subject except 
that subject's own, and sees difficulty in carrying his subject in his 
head. 

* With thanks for my copy of the above-mentioned.' 

From the SAME to the SAME. 



* 7, CAMDEN-STREET, CAMDEN TOWN, 
'April 6, 1850. 



' I cite you to tell me who the abstracts are for, with the f ollow- 
ing theorem: 

[He then takes the equations of a line x& + y,y + 1 = 0, :m<l 
calls this line and the point P, whose co-ordinates are #1, y\ 
relatives. He next takes the equations of two conies such that 
the coefficients of the second are the same as those in the tan- 



Professor Augustus De Morgan. 285 

gential equation of the first, and mentions that those of the first 
are similarly related to the co-efficients of the second.] 

I Or the connexion of the co-efficients is one of reciprocity. 

f Moreover, if any point whatsoever be taken in the plane of 
the two curves, say P ; and if the curves be called (v) and ( F), 

' The polar (point) of the relative (line) of P, derived from (v) 
is the same as the relative (point) of the polar (line) of P, derived 
from ( F). And (v)ice (V)ersa. 

* And this last theorem contains, and is contained in, the rela- 
tion of the co-efficients mentioned above. 

I 1 was looking to day over my Cambridge and D UBLIN Mathe- 
matical Journals, making them up for binding. I really think it 
incumbent upon you to write to the editor, in the name of the Irish 
nation, to know what he means. Perhaps you would prefer to call 
a council of Irish mathematicians first, to discuss what steps should 
be taken. 

1 You will observe that, from the moment that the Dublin 
Journal was joined with the Cambridge one, a certain Greek motto 
was put on the title-page. Now under this motto lurks a sly and 
insidious piece of Saxon impertinence (it is well O'Connell is not 
alive). The motto is, Auwv ovo/uaTwv juop^rj /zm which can mean 
nothing but " Of our two names Murphy is one." This is too bad, 
really : who would have suspected a quiet man like Thomson of 
such a thing ? ' 

From SIR W. B. HAMILTON to A. DE MORGAN. 

* OBSERVATORY, Tuesday night, April 9, 1850. 

' The theorem you have been so good as to communicate, in a 
letter received this evening, seems to me (on, as I own, too short 
an examination), to be included in the following, which I suppose 
is known : 

6 " If two conies (A) and (C), be polar reciprocals of each other, 
with respect to another conic (B) ; then whatever point P may be 
taken in their common plane, the (A) pole of its (B) polar coin- 
cides with the (B) pole of its (C) polar." 

* In fact, for three concentric circles, with radii in geometrical 
progression, this theorem is obvious. And I think that a person 
accustomed to the modern extensions of geometrical results could 



286 Correspondence between Sir W. Rowan Hamilton and 

scarcely doubt, after seeing this, that the theorem is generally 
true. 

4 Now let the conic (A) be your curve (), and let the conic (B) 
be the imaginary unit circle round the origin, 

x* + y + 1 = ; 

the conic (C) will then become your other curve (F). 

* Thus substituting (v) and (F) for (A) and ((7), and observing 
that the (B] pole of a line is what you call its relative point, while 
the (B) polar of a point is what you call its relative line, your 
theorem emerges under the form that 

1 The (v) pole of the relative line of any assumed point P in the 
plane coincides with the relative point of the ( F) polar of the same 
assumed point P ; the conies (v) and ( F) being of course (as you 
notice) interchangeable. 

* I shall be glad to know (at your leisure) whether this view of 
your theorem has occurred to you, and whether you approve of it.' 

From A. DE MORGAN to SIR W. B. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 April 14, 1850. 

1 Be it known unto you that I have discovered that you and 
the other Sir W. H. are reciprocal polars with respect to me (intel- 
lectually and morally, for the Scotch baronet is a polar bear, and you, 
I was going to say, are a polar gentleman, only I thought perhaps 
you might go and say I called you an Esquimaux). The intellec- 
tual polarity is of the kind $^x = - x. When I send a bit of in- 
vestigation to Edinburgh, the W. H. of that ilk says I took it 
from him. When I send you one, you take it from me, general i/.e 
it at a glance, bestow it thus generalized upon society at large, and 
make me the second discoverer of a known theorem. He cuts my 
legs off; you make a pair of legs grow out of my head, and turn 
me upside down to stand upon them. His process after yours 
gives ^x = - x. Reciprocal polarity the last and most agreeal>K> ; 
your process involves no writing of pamphlets. 

' 1 believe you have hit it, but I cannot find that your tin- 
is known. I have not yet demonstrated it ; but, from your instance 



Professor A u gust us De Morga n. 287 

and mine, it may be that this theorem is only true when in the 
curve of reference 

coefficient of if = coefficient of x*. 
Poncelet has no theorem of the class that I can find.' 

From the SAME to the SAME. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 April 14, 1850. 

' At you again. I take it that the theorem you gave me in 
involved in the very definition of reciprocal polars. 

[This he establishes by stating the now well-known principles 
of polar reciprocation.] 

1 N. B. It is not necessary that the reciprocal polars should be 
of the second degree. Now I very much doubt all this being 
found anywhere.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, April 18, 1850. 

* I abstained yesterday from reading your note No. 2, as 
intending to turn on the quaternions, and prove (or disprove) by 
them the theorem which a former letter of yours had suggested to 
me. The example of the concentric circles, although encouraging, 
did not upon consideration appear to me decisive, because, as you 
know, any two such circles in one plane are considered to have 
double contact at infinity. I therefore attacked, this morning, the 
analogous question respecting surfaces of the second order, using 
my own calculus, which adapts itself, at least with me, more easily 
to surfaces than to curves ; and I soon had the satisfaction of 
seeing the theorem confirmed, at least for the case when the inter- 
mediate surface is a sphere, which case appears to me to be quite 
general enough, and by no means confined to yielding a mere 
induction or probability, however strong. In fact we may always, 
by real or imaginary deformation, reduce any one central surface of 
the second order to become a real sphere of a given radius round ;i 
given point as centre. Only what had before been real may then 



288 Correspondence between Sir W. Rowan Hamilton and 



become imaginary, and we may have to consider imaginary points 
upon the real sphere, and I infer then that, in space, " If any two- 
surfaces of the second order, A and C, be reciprocal polars of each 
other with respect to a third surface of the same order B, thfr 
A-pole of the B-polar of any point P whatever coincides with 
the B-pole of the C-polar of that point." But my mathematical 
learning is far too slender to enable me to pronounce with any 
confidence that this is a new result. 

' Since satisfying myself of its truth, I have read your note 
No. 2, which seems very clear and satisfactory. Only I do not 
quite understand how you propose to extend the theorem to curves 
beyond the second order, on account of there being then several 
tangents, and no one definite rectilinear polar of a point ; but pro- 
bably you have some way of fixing what seems here unfixed. 
Perhaps you use conies instead of right lines, in passing to curves 
of the third degree. It is fair to remark that for conies your 
theorem may be regarded as virtually including the one which it 
suggested to me, and not merely as being itself included therein ; 
for the imaginary unit-circle # 2 + y 2 4- 1 = 0, which I called the 
conic (B), may be made by deformation to represent any other 
conic. My merit in the matter, if any, is very small ; and I shall 
be happy, if I ever print my result in a Magazine or Journal, to 
acknowledge that it was suggested by a communication from you. 
I wished to show you that I am not always so stupid, indolent, or 
busy, as to lay aside unstudied the hints with which you occasion- 
ally favour me ; and I took the opportunity to let my eldest boy 
see me go through some work with co-ordinates, for the purpose of 
verifying your theorem, and confirming the guess that it gave rise 
to. As to the subsequent extension to surfaces, I shall just copy 
here a formula or two of such co-ordinate work ; but assure you 
that they are (in my case) mere translations from a previous, more 
general, and, as I think, generality being allowed for, more siniplo 
investigation with quaternions. 

' Let, then, the first two surfaces be 

(A) <r (x - /)* + b~* (y - my + <r 2 (z - iif = 1, 
and 

(B) a' + if + s' = 1 : 

let also the point XFZbe the B-pole of the A-polar of any assumed 



Professor A u gust us De Morgan . 2 89 

point jcyz , we shall have the relations 

a~ 2 (x-l) b~* (y - m) f*(z-n) 



= 1 + or*- 1 (x - 1) + b~ z m (y - m} + c~ z n (z - n] 
= .'. (1-lX-mY-nZ}- 1 ', 

whence the equation of the reciprocal polar of (A), taken with re- 
spect to (B) is 

(C) a*X z -f V Y* 4 c*Z* = (1 - IX - m Y- nZ}\ 

1 If now we seek the B-pole, xyz, of the C-polar of an arbitrary 
point of space, XYZ, we are reconducted to the same relations as 
before ; which proves the theorem, at least for the two assumed 
surfaces (A) and (B) above, and therefore, by deformation, for any 
two surfaces of the second order. 

* Hoping to write soon again, I remain, in the meanwhile,' &c. 

[A letter of Sir W. E. Ii., dated August 26, 1850, replies to 
one of De Morgan's on a logical subject. It is not here printed, 
as not easily intelligible without the letter to which it refers, and 
which I have not been able to discover. E. P. Gr.] 

From the SAME to the SAME. 
[FROM A DRAFT.] 

* OBSERVATORY, October 19, 1850. 

'I.I am in the mood for writing to you something about the 
first principles and fundamental rules of the Quaternions considered 
as a Calculus, and that I may diminish the risk of boring you, and 
have a chance of extracting your comments with the least possible 
trouble to yourself, I shall take as my text-book your very valu- 
able little volume on Trigonometry and Double Algebra (London, 
1849), which you were so good as to send me at the time of its 
publication. 

* 2. The Calculus of Quaternions may perhaps be defined to be 
that extension of ordinary Algebra in which the square roots of 
negative numbers are interpreted as denoting vectors, or directed 
right lines in tridimeusional space ; and in which such lines are 

VOL. in. u 



2go Correspondence between Sir W. Rowan Hamilton and 

supposed to be not generally commutative with each other as 

factors in multiplication. They may, however, be so commutative 

as factors, namely, when they are parallel to one common line; 

and thus the Calculus of Quaternions includes all ordinary Algebra. 

' 3. A general theorem of (ordinary) Algebra is not necessarily 

(nor even usually) applicable, without modification, to quaternions ; 

but every general theorem of quaternions is true for ordinary 

Algebra. For example, it is not true generally in quaternions 

that 



but it is true in algebra, as well as in quaternions, that 

(r + qf = r* + rq + qr + r z . 

' 4. Again, it is not generally true in quaternions that 
d . <f = 2q . dq, 

when d is the mark of differentiation ; but it is true in algebra, as 
in quaternions, that 

d . q z = q . dq + dq . q. 

1 5. Again, we must not write in quaternions, as in algebra, 

d . q~ l = - q-*dq ; 

but we may write, in algebra, as in quaternions, 

d . q~ l = q~ l dq q' 1 . 

In general it may be said that the differential and integral calculus 
of quaternions remains to be formed ; although a few things have 
been published by myself on the subject, and on certain analogues 
to Variations and Partial Differentials, with some applications and 
some solutions. 

'6. Although my "vectors" are all square roots of negative 
numbers, I yet am in the habit of avoiding the use of the symbol 
^/- 1 ; not as at all uninterpretable in my system, but as being too 
easily interpretable ; as being in fact (in it) indeterminate and ray tie. 
Every unit-vector in tridimensional space is with me equally entitled 
to be described by the symbol ^- 1 ; I therefore usually avoid to 
use that symbol as denoting any on* such unit- vector ; and emj 
commonly, for any Articular vector, whether equal or not (Mjual to 



Professor Augustus De Morgan. 291 

unity in length, some special symbol, namely (such is my custom) 
some one small Greek letter, selected according to what may seem 
the [most] convenient in the case, and having accents or indices 
annexed, or not annexed, according to what seems convenient. 
Thus, a, /3, 7, ... ai, |3', 7" . . . are all with me (often) used as 
symbols of vectors ; and then it is to be understood that I regard 
the square of each as a negative number.' 

From the SAME to the SAME. 

'OBSERVATORY, November 26, 1851. 

* I have often been so abominably lazy as to acknowledging 
your kind attentions that on this one occasion, at least, I resolve 
not to lose a post. Let me then thank you for your Paper, on 
" Recent Discoveries on the Invention of Fluxions," received by 
post this morning. 

' I think that you do our species a service, by assisting to vindi- 
cate, from the charge of plagiarism, the memory of a great man 
[Leibnitz]. For my own part, I find it difficult to believe in any 
instance of such a thing, it is so pleasant to confess an obligation. 
Yet with the usual inconsistency of human nature I have (I own 
it with regret) at times suspected that / have been plagiarised 
from, by one (not very high) contemporary not you, nor any one 
/ike you. But I deeply distrust such distrust. 

* My book on the Quaternions is advancing rapidly I have 
just been correcting the slip 2 r 3, which will bring it somewhat 
beyond 440 octavo pages. I first aimed at 200, but shall now 
congratulate myself if I get off under 500 pages.* You should be 
most welcome to copies of all the sheets hitherto printed, if I 
fancied that you would accept them, in the present state of the 
publication. In fact I should like to send them, but think it not 
quite fair to force what may be thought a confidence on anyone. 

* You might do me a material service, if you chose, with little 
or no distraction to yourself. I mean that you are so much better 
acquainted than I am with the history of science, ancient and 
modern, that it would cost you little or no trouble, while it would 
be hard work and new work to me, to put the finger on the books 
which ought to be acknowledged, as preceding the invention of 

* The number, including the preface, did actually reach 888. 
U 2 



292 Correspondence between Sir W. Rowan Hamilton and 

the quaternions, and as having any sort of connexion therewith. 
Sometimes it has occurred to me to ask your leave to reprint your 
list of references, prefixed to your own remarkable book on Double 
Algebra ; but again, I have thought that this might seem to be 
assuming to myself the merit of a certain amount of research to 
which I have no claim ; for there are several of the works, referred 
to in your list, which I have never even seen. Indeed it is as 
notorious as public confessions of mine can have made it, that a 
degree of familiarity with what you have happily named Double 
Algebra, preceded in my own mind, by several years, the thought 
of quaternions andsete; although the conception of such "sets,'* 
suggested by views about time, occurred to me at least as early as 
1834, and was publicly written of by me in 1835. But, on the 
whole, I have several times thought, and Charles Graves appeared 
to agree with me when I once mentioned it to him, that the rela- 
tion between Double Algebra and the Quaternions is rather one of 
contrast than of resemblance, although it is undoubtedly possible, 
by some not difficult modifications, which I should be happy to 
submit to you, to exhibit the quaternions as including the results 
of Double Algebra. A quaternion may always be reduced to the 
form of a couple, 

a + y/Ii b. 

But in my nearly finished work, which will, however, cost me 
several weeks of hard work still, I have chosen, for the sake of what 
seemed clearness, to take a more purely geometrical view of the 
whole matter, with a continual reference to space of three dimen- 
sions. And it is chiefly with reference to such a view, that I desire 
to be furnished with references to persons who may have precede! 
me. But if, either on this or any other aspect of the matter, \<>u 
choose to favour me with any remarks, I shall very gladly rec 
them; and if you choose, or permit, it would give me pleasur 
prefix, or append, to my book, any letter of yours on the subject.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDKN-STREET, CAMDEN Tow .\ , 
'November 26, 1851. 

* I have seen one sheet which your son brought me and a vt-rv 
likely youth he seems I think you will find him rnako his \\ M y 
if one dare judge by a few minutes' conversation. 



Professor Augustus De Morgan. 293 

' I beg you will send me the part printed, without scruple. 
Tli ore is a pleasure in reading while anything that strikes may do 
service ; it is the reviewer's feeling Christianized. 

' There is no danger of my being in any of your thoughts, in 
any manner which may oblige me to take care what I do. For I 
have, ever since I published on triple algebra, had a kind of re- 
pugnance to letting the country beyond double algebra occupy 
my thoughts, and I am quite bare of everything except casual 
reading on quaternions from your pen. I hit accidentally upon 
what I believe to be the reason of this very curious and with me 
unusual, indeed unique recoiling from a subject I had once 
thought about. Taking up a copy of my Paper on Triplets I saw 
the date, and it reminded me that while I was thinking about it 
poor F. Baily was lying on his death-bed ; and, as I used to walk 
down day after day to hear that he was growing worse and worse 
I had the triple algebra in my thoughts ; and I believe that after 
the Paper was printed I used to remember it with a kind of disgust. 

' Another Sir W. Hamilton once offered me communication of 
his unpublished speculations, and we had a row about it. But 
there is no fear here. First, because I am not on the same sub- 
ject; secondly, because you are not a controversialist by habit, 
and have too many novelties in your head to be mono some- 
thing, or very near it about one. The history of Sir W. H.'s 
mind on the quantification of the predicate would be a curious 
thing. 

( Assuredly the quaternions must contain the double algebra. 
Your couplets of time must have suggested triplets and n-lets. 
(N. B. What are ^/- 1-lets.) But you can hardly have failed 
to look at the question in connexion with double algebra not as 
time at the moment when you thought of the triplets. I should 
suy j9flce Graves and yourself both that both contrasts and 
analogies must have occupied your mind. In fact, we may ask, 
do contrasts ever strike except in connexion with analogies? We 
contrast green and red : do we ever think of the contrast between 
green and loud ? certainly not, except when we happen to contrast 
sight and hearing perceptions both. Differences are, I think, not 
thought of as other than the separators of. species under one genus. 
And there is logic for you. 

* I should be very glad, if you would tell me in confidence of 



294 Correspondence between Sir W. Rowan Hamilton and 

whom it passes through your mind that he has plagiarised you. I 
think I might perhaps remove the impression, or point out some- 
thing still stronger in some person above suspicion. I have thought 
a good deal ahout plagiarism guilty and innocent. There is a 
kind of plagiarism which no one can guard against. It is the Uyht 
which another person's results throw upon his own previous thoughts, 
without taking an atom of his wick or his oil. 

* I see that by a most extraordinary f orgetf ulness I have not 
mentioned your time-paper in the list you refer to. I can only 
account for this by supposing that I made sure I could not forget 
it, and made no note, or perhaps I could think of nothing but the 
geometrical view. 

' Wheatstone (who has a great love for pure speculation) is 
very strong upon what the Germans have been doing. When I 
get your sheets I shall apply to him to know what he can tell me. 

' P. S. Unfortunately one cannot clear Leibnitz without incul- 
pating others. I think myself that plagiarism is very uncommon 
among really great men. But I have my doubts as to Laplace.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

< ROYAL IBISH ACADEMY, December 1, 1851. 

* I received with great pleasure your note, by a late post on 
Saturday, when I was about to start for an evening meeting of the 
R. I. A., and now write these few lines to acknowledge it, at the 
Council table of the same body, after the other Members have 
gone. 

* At this moment I only send you as an instalment the first two 
sheets, and you will see that the printing was begun so long ago as 
1848, though it is only lately that I have been really active at the 
work. 

*I am aiming at publication early next year, but shall prixo 
very much any criticisms from you, and any hints respecting 
anticipations, or even concurrent or later investigations, which 
have any connexion with the subject. Something (not much) I 
do know of what has been done abroad, which I hope to notice in 
preface, notes, or appendix. My knowledge of it is too slight, and 
came too late, to enable me with any convenience, or indeed pro- 
priety, to speak of it in the text; it is, for instance, pretty recently 



Professor A ugustus De Morga n. 295 

(long after my printing was begun), that I have heard anything 
of an employment like my own, of B - A in geometry. To me 
the thought of using that symbol was familiar, I suppose twenty 
years ago. More than sixteen years ago I published the analogous 
iiM' of it, as denoting what I called a time-step, as it is in the quater- 
nions a space-step (from A to B). So little am I sore on points of 
that sort, though indeed it would be hard for me to have that 
feeling towards you, that I never missed my name, or the " Pure 
Time," from your list of references (which is not just now at my 
hand), nor do I think that you were bound to refer to it, even if it 
had been in your remembrance. 

' I have every confidence in Wheatstone. Act as you think fit 
respecting him. 

* How do you advise me to send the other sheets ? ' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEN TOWN, 
' December 4, 1851. 

* Being deep in such a sublunary thing as a life assurance 
report, I have no time at this moment to do more than acknow- 
ledge. Next week I shall set to reading being (this word to pre- 
vent ambiguity) yours truly.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 
[FROM A DRAFT.]* 

' OBSERVATORY, December 8, 1851. 

* If you allow me to continue to correspond with you on the 
subject of my forthcoming work, you must make up your mind to 
my being often intensely egotistical, especially at present. Ee- 
member that as far as separate publication goes, I am a virgin 
author \ feel quite conscious and embarrassed, although having 
been in some degree before the mathematical world since my boy- 
hood. It is almost exactly twenty-seven years since Dr. Brinkley 
laid my first paper (and an elaborate one it was ; it is in my posses- 
sion) on a general theory of Caustics before the Council of the 
Eoyal Irish Academy. 

* It would appear that this letter was not sent. 



296 Correspondence between Sir W. Rowan Hamilton and 

'You would do me no great good by criticising and I can 
conceive your feeling a temptation to do so the metaphysics of the 
early articles at present. I know that I have not done justice even to 
my own views in that direction, much less to existing philosophy : 
cW liberavi animam meam. I could not bring myself to enter on 
the subject without some such introductory remarks, and must 
only hope that while thus imperfectly recording some moods or 
frames of mind, which have really influenced myself in mathema- 
tical speculations of the class we are now considering, I shall not 
have materially embarrassed the path of any student who will 
have even a moderate degree of patience to wait till he sees how I 
use the notions with which I profess to set out. At one time I 
read a good deal of Kant's works in the Grerman, besides portions 
of Plato in the Greek : it is one of my hopes to resume, at what 
may be called leisure hours, some of my old studies of that kind, 
and to combine with them the reading of some other and more 
Aristotelian than Platonic works including the " Formal Logic ? " 
although my own personal temperament of mind is far more 
Platonic than Aristotelian. Don't be so malicious as to quote the 
Malim cum Platone errare, however applicable you may think it to 
be ; and let me tell you that when I was a boy at College I acquired 
some undergraduate renown by a short proof (which I have totally 
forgotten, and which would at all events have been since super- 
seded by one of Mr. Boole's), that in no legitimate syllogism can 
the conclusion change place with either of the premises. 

* At Oxford, in 1847, I showed as your moon to Mr. Hallara, 
the historian, reflecting on him at breakfast in Christ Church your 
illustration about the white cravats and white waistcoats (was that 
it ?) having each a majority in some party, some member of whu-h 
was thence inferred to have icorn loth. The earth revolved on its 
axis ; and Mr. Hallam told me the next morning, at our next 
breakfast, that he was convinced. It is only honest to add that 1 
passed an evening in company with my namesake in Edinburgh 
in 1850, and thought him an interesting person, although it was 
often very difficult, from his physical infirmities, to collect distinctly 
what he said/ 






Professor Augustus DC Morgan. 297 

Front the SAME to the SAME. 

4 OBSEIIVATORY, December 10, 18ol. 

* Thanks for your line of acknowledgment of my first two 
sheets. You have, I hope, received three others since. 

' I have not /w// 1 answered your former note, but shall return 
to subjects contained in it plagiarism and all that. As to myself, 
I am sure that I must have often reproduced things which I had 
read long before, without being able to identify them as belonging 
to other persons. However, in the present case, as regards my 
work in hand, you will see, and I must try to make it plain to 
others, that I do not design it as a treatise on algebra, nor on 
geometry, nor on the philosophy of mathematical science, nor even 
on the interpretation of the symbol */- 1 (which, in fact, I rarely 
use) ; but simply as an introductory treatise on the quaternions, 
designed to make quite clear the meanings of the chief notations of 
my calculus, and the truth of the chief principles which I employ, 
with respect to the combinations of its peculiar symbols ; the whole 
being viewed, on this occasion, with express and almost exclusive 
reference to geometry of three dimensions, although illustrated 
from astronomy, and indeed partly from mechanics. Unless it 
can be shown that I have been anticipated in the conception of the 
quaternion, especially as connected with such geometry, or in the 
laws of combination of my i 9 /, k (i 2 =j z = k 2 = ijk = 1), not to speak 
of the notations K, S, T, U, F, which might at need be dispensed 
with, I shall with great equanimity see claimed any subordinate or 
illustrative results ; though it is true that I do, at present, suppose 
many of the results of the calculus to be new. 

* Even if the ijk themselves should at last go overboard from 
my own bark into the general sea of science a very bad metaphor ! 
or (better) if, on farther search through my own little island, I 
shall find planted some former flag, I trust that I should still lose 
no jot of heart or hope. 

' Non ego, cum scribo, si forte quid aptius exit, 
Quando hsec rara avis est, si quid tamen aptius exit, 
Laudari metuam : neque enim mihi cornea fibra est : 
Sed recti finemque extremumque esse recuso 
Euge tuum et Belle. 

'As to my terminology, I took pains with it, but am awaro 



298 Correspondence between Sir W. Rowan Hamilton and 

that plausible, perhaps just, objections may be made. I might 
have shrunk from adopting the word "scalar," which I did (I 
think) in 1844, if I could have known that you were to use that 
word in so different and in so important a sense, at the close of 
your Double Algebra. It was not easy to foresee that you would 
establish such formulae as 

(* + *) x ,, ( c + <0 = a * c + a x " d + b x // c + bx ,, d '> 
the common value here being 

y*{\(\a\b)\(\c\d)}. 

I don't want to explain away my obligations to Double Algebra 
in the filiation of ideas, it was the parent, or grand-parent, of the 
Quaternions an undutiful chap, who complains that its old dad 
does not go fast enough ahead ! In fact, as respects geometry at 
least, my expectations from Double Algebra are very limited. 
Even within the field of the two dimensions, it has in my eyes the 
fatal fault of leaving no imaginaries, or at least no impossibles : an 
odd complaint to make ! but there really are unrealities in geometry, 
such as the intersections of a circle with a wholly external right 
line, and these must be dealt with. Now my quaternions deal 
with all such with ease and profit, at least my biquaternions do so, 
a biquaternion being 



where v/- 1 is the old symbol, and q, q' are quaternions. They 
reproduce, for instance, Poncelet's ideal sections, and lead to many 
new and analogous results.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 December 15, 1851. 

* I have received up to page 112, and have pretty nearly 
tln-m, but am too busy to write this week. However, I think you 
need not be afraid of missing any acknowledgments you ought to 
make. Next week I will discuss this topic.' 



Professor Augustus De Morgan. 299 



From SIR W. E. HAMILTON to PROFKSSOR DE MORGAN. 

OBSERVATORY, December 16, 1851. 

* I intend to have a set of notes, at the end, with no other 
arrangement than the numbering of the articles of the text, so that 
there will be the greatest ease and freedom for inserting any remarks 
which you may think fit to make, and may allow me to insert, with 
your initials, or name in full, if to both of us that course shall seem 
good, as I can scarcely fancy my own objecting to it. 

* I am quite aware of faults, but do not feel, or rather do not 
judge, that longer time spent on consideration would have made 
the work much less faulty. And as to history, I despair of writing 
anything worth notice on that, within the remaining time, except 
so far as I may be assisted by friends. Perhaps I shall be driven 
to make one sweeping and true declaration of want of preparation 
on that head/ 

From PROFESSOR DE MORGAN to SIR W. R. HAMILTON. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
'December 17, 18ul. 

' I am glad you have adjourned till February. Do not be too 
sure even of that month. 

* I will set you at rest about the history in a day or two/ 

From SIR W. E. HAMILTON to A. DE MORGAN. 

1 OBSERVATORY, December 18, 1851. 

' Of course you must be right in saying that the sheets of my 
Lectures, which you had received when you last wrote, went no 
farther than page 112. But I had fancied, and had even written 
it down as a memorandum in one of my many manuscript-books, 
from recollection, however, when so writing it, that my sealed 
packet of the 10th contained the three sheets, A, H, I, extending 
to page 128. It will console me for the blunder so committed in 
my memorandum made from recollection of a packet dispatched 
in haste, if I shall find myself to have aho blundered in another 
part of the said memorandum, and not in my letter to yourself. 
I refer to my quotation from Persius one that, like some others 



300 Correspondence between Sir IV. Rowan Hamilton and 

from the same stiff, stoical, uncyrenaic, and non-Horatian writer, 
although he had the taste or sense to admire Flaccus, impressed 
itself long ago on my own somewhat stoical mind, such as the 
" Neo te quaesiveris extra," on which indeed I plead guilty to 
having once written a sonnet. Was I then such an ass as to write 

" Euge tuum atque Belle " ? 

instead of et. After dashing off my letter to you, I went out to 
walk in my garden which was once Brinkley's, and has had its 
walks trod by many illustrious visitors ; Wordsworth was once a 
guest of mine, and I have a " Wordsworth's Walk," and the said 
garden led me long ago into the perpetration of a couple of other 
sonnets and while I was so walking, the enormity of my offence 
against Prosody flashed across me, if indeed I did commit it, by 
writing the aforesaid Atque. What could De Morgan think of me ! 
To be sure, he says, in his "Formal Logic," that he is not scholar 
enough to give a decided opinion on the meaning of a certain 
Greek verb; but that is just one of the few points on which I 
won't take his own icord. 

* By the way, I picked up yesterday, for the splendid sum of 
one shilling, a work almost exactly as old as myself, and which 
will be, I think, a real treat to you. I have already written in it 
an inscription to the author of the " Formal Logic," in the pages of 
which book you somewhere avow that you have been unable to 
procure the work I speak of ; the beauty of the business being 
that you did not know so much as how to refer to it. Pro- 
bably you don't even know its name : but / made sure of there 
being a correct and proper title-page (1805). Enjoying the plea- 
sure of tantalising you, and of, for once in my life, knowing more 
of bibliography than yourself, I remain, &c.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 December 20, 1851. 

* Your memorandum is right and wrong to your liking. Y<>u 
did send up to p. 128, and you did not write atque for ct. And as 
to my detecting it, I should certainly have known that nfijiir 
was not a dactyl, because I remember about long by position (a 
right the parson too often takes in the pulpit) ; but if it had boon 



Professor Augustus De Morgan . 301 

one of your doubtful cases, I never could have trusted my own 
memory. What I say about my scholarship is but too true. In 
middle Latin I am a fluent reader, and that is the best I can say. 
At school I could do what I liked ; but as I also would do what I 
liked, I turned off to mathematics. 

* I shall be very much obliged to you for the book. I remember 
no place where I have mentioned a book I could not name, except 
in the appendix, where I say I have heard some books (not book), 
used in the Irish Colleges, make the rules of syllogism expressly 
dependent on the exclusion of contraries. Graves was my informant, 
but he afterwards told me that he did not so much refer to books, 
as to cuts (as he called them), common in college, meaning pro- 
blems. But I shall be glad to see a book of your date to watch 
this point. 

* On Monday, I think, I shall write on your question of priority. 
Wheatstone sent me a list of some books to day. 

* And so you have not given up sonneteering. Did you ever 
read the poem called the " Loves of the Triangles " ? or Bosco- 
vich's poem on Eclipses ? or St. Prosper on Grace ? the only 
poem the Jesuits, who educated D'Alembert, would recommend 
him to read ; they said poetry dried up the heart. As you are a 
namtur-non-fit poet, you have a right to your inspirations.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

'OBSEB.VATOEY, December 23, 1851. 

* I send three more sheets, and regret to say that I am not now 
sure (on reflection) that you will care much for the logical book which 
I picked up for you, although I still intend to forward it, as a little 
mark of good will. It ran in my head that you had stated, in one 
of the notes to the " Formal Logic," that you had heard of the ex- 
istence of a work used in Dublin, without being able to procure it, 
which work contained some coincidence with your views, on some 
important point, I forget what. And I remember well that when 
a boy I had read carefully Walker's Logic, as we used to call it, 
but in fact a Commentary on a Compendium by Murray, which 
latter was a poor concern, and although nominally a text-book here 
at the time, was generally considered to have little other value 
than what the comment gave it. And I knew that Walker John 



3O2 Correspondence between Sir W. Rowan Hamilton and 

"Walker, founder of a religious sect, editor of Lucian and Livy, 
author of a sound book on the Philosophy of Arithmetic, and on 
the Elements of Algebra, a learned and good man, whatever / 
may think of his leaving the Church of England, which it is fair 
to say he did at great personal loss had laid much stress on the 
possibility of treating negative propositions as affirmative ones, and 
of converting them as such. For instance, his not-mastiffs, not-a-dog, 
rang still in my ear. I suppose Walker conceived that the Dublin 
youths whom he addressed were sons of sporting sires, and that a 
talk about greyhounds was likely to allure the gaudent[_es~\ eguis 
canibusque. But I am by no means certain that this was the point 
(if there was any) on which you had been told of some concurrence 
with your views. 

' P. S. The edition is the first, and I believe now scarce- 
Dublin, 1805. I took care to see that there was a title-page, but 
cannot copy it now. 

' 2nd P. S. I was reading for some hours, before breakfast this 
morning, Warren's book, and yours, on /- 1, and you have nar- 
rowly escaped a letter on that subject.' 

From the SA.ME to the SAME. 

' OBSERVATORY, December 24, 1851. 

* I am not such an inveterate sonneteer as you seem to suspect 
me of being, at least as to quantity, for I only composed four this 
year, and they were called forth by the death of the late Lord 
Northampton, with whom I had long enjoyed what might be 
called, and what in fact he did call, intimacy ; while yet we were 
sufficiently far separated, by rank and fortune, and even by our 
living habitually in different parts of the world, to prevent that 
crushing sensation of disaster, which has hindered me from ever 
writing any verses on the death of any blood-relation of my own ; 
were it not for which sensation, I might have felt inclined to 
attempt some elegiac record of my poet sister, Eliza Mary 
Hamilton, who died in my arms last May. 

* I send you the only one of the three old sonnets, monti> 

in a former note, which even J can think worth copying ; ami 
Apropos to the chief subject of our present correspondence, I throw 
in a sonnet on " The Tetraotys," occasioned by the recollection, in 



1 ) i 'ofessor A ugnstus De Morga u. 3 03 

1846, of conversations with Sir John Herschel, enjoyed that y-;ir 
at Collingwood. On my showing tlie lines afterwards to a poetical 
and philosophical friend, since dead, Professor William Archer 
Butler of Dublin, he said to me, "I see clearly now that your 
Quaternions are a gross plagiarism from Pythagoras." A curious 
acknowledgment would not this be, to be included among those with 
which you and Wheatstone will supply me ? I have no depot of 
sheets here, but send z, &c. 

* I wrote something about the logic yesterday.' 

From the SAME to the SAME. 

' OBSEKVATORY, Christmas Eve, 1851. 

* To give myself a chance of forgetting the Quaternions to- 
morrow, I desired the messenger, who was to post my note of 
to-day, with the sheets z, 2 A, 2 B, to procure, what I now enclose, 
and what completes my book so far as hitherto printed off, although 
most of sheet 2 H is in type, the three remaining sheets, x, Y, and 
2 G, from the store with my printers in Dublin. I am ashamed to 
say, " from their depot" or from my depot with them, after using that 
word, as I now remember that I did, this morning, and omitting 
the circumflex accent. Certainly I make no pretension to remem- 
bering the French accents generally ; an Oxonian would shudder 
to be told, what is too true, that I am constantly forgetting the 
Greek ones. But in this particular case, if I had not written in 
great haste, I ought to have recollected the obvious etymology, 
depositum, depost, depot. I have, or had, a charming old book of 
fairy tales, and am sure that I have read other old French books, 
in which one constantly met such spellings as estre, qu'ilfust, and 
soforth, since changed, as every one knows, to etre,fut : fete was 

, &c. ; though I won't assert that the rule is universal. 

' My own spelling, in the book in hand, has perhaps often been 
capricious. I don't intend to bind myself by it in any future 
work, though for the moment I have aimed to be at least consistent, 
and even this humble aim has not been always attained, in the case 
of the words farther and farther. I scarcely ever look into an 
English Dictionary, and for many years had no Latin one here 
not till my boys required it ; but when I was a boy myself, what- 
ever language I was studying, I used to make it a rule, never to 



304 Correspondence between Sir W. Rowan Hamilton and 

grudge looking out for a word in the Lexicon, or other word-book ; 
which of course made it more possible for me habitually to dispense 
with them afterwards.' 

From A. DE MORGAN to SIR "W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 

' December 27, 1851. 

' I have never fairly got time to sit down for a long letter till 
this evening ; but I will, D. V., settle the plagiary question before 
I finish it. Various packets the last received this morning give 
me everything up to p. 464. To make a clear stage. First, thank 
you for the sonnets, the one about Tetractys is quite professional 
and happily worded. I trust the last line of the one to the Garden 
is only poetry, and that you do not set up for a really unhappy 
man.* There is every right to tell downright falsehoods in poetry : 
rhyme and reason are what my logical terminology calls sub-con- 
traries, or as I freely construe Horace Semper fuit cequa potesta* 
it is quite impossible to say quidlibet audendi which are the 
greatest liars pictoribus atque poetis painters or poets. 

* The book on logic will not be a bit the less useful, because I 
do not refer to any specific book. If I can establish by positive 
evidence that the idea of contraries affecting the laws of syllogism 
was floating about in university problems, and by negative evidence 
that it was never introduced into the books, it will be a fact worth 
noting one little instance of the fear of going beyond or out of 
Aristotle, which prevails to this day. 

' John Walker was the very man to be an exception. I did 
not know him, but he lived in Camden-street years ago, within a 
door or two of where I now live. I have heard the people In TO 
speak very highly of his scholarship ; and I know his arithmetic 
somewhat, and his geometry better. The last is a very powerful 
book. I shall be very glad to have his Logic, or his Notes. 1 
books are not much on the stalls : those who have them keep them. 

* As to depot, I hold that the word has become vernacular :nid 
has lost its accent. You say you have forgotten the Greek accents. 



Some transient vision, or some conflict brief 
'Twixt the intruder Joy and dweller Gru-t'. 1 

Supra, vol. ii., p. 



Professor Augustus De Morgan. 305 

I never knew them, at least in school teaching. I do not believe 
in them. I have heard it suspected that they are the work of the 
time when pronunciation began to depart from quantity, and it 
became necessary to mark the mode of departure. 

* Now as to the quaternions. Your Paper on pure time was 
read in November, 1833,* and the essay on quaternions ten 
years after (November, 1843). 

' Peacock's report on analysis was published in 1834. It is so 
full on the foundation of algebra, and Peacock is so much master 
of the bibliography of that time, that I take it as beyond question 
the works which he omits may be considered as out of your reach, 
unless you have special reason to know the contrary in any case or 
cases. 

* Wheatstone mentions the Theories Residuorum Biquadraticorum 
commentatio secunda, by Gauss, Gottingen, 1832 ; and says that it 
is a notice in this work which has avowedly called the attention of 
all the subsequent writers in Germany to this subject. 

'It seems, then, that about 1831-33 there was a revival, as the 
Americans call it, of pure algebraical speculation that you have a 
share in it as the author of the Paper on pure time. 

' It seems to me that all the generalities of the Paper on pure 
time are in the fullest sense incapable of being transferred else- 
where, unless you can do it yourself. The subject was not at that 
time in vogue. Mr. "Warren's books (whether you saw them or 
not) do not treat of general interpretation, or d priori search after 
significance, but specifically treat of one aspect of */- 1, and ex- 
pressly reject a part of algebra as not interpretable (e ev ~ 1 , for 
instance). 

* I believe this Paper on pure time puts you beyond the neces- 
sity of looking elsewhere upon the whole general idea of algebra. 
The allusion to triplets in the last sentence shows you prepared, in 
any system which masters a + fty/-!, to ask after ai+bj + ck. I 
should think that some of the fundamental points may have struck 
others it is so in everything. In my first memoir on the foun- 
dation of algebra I mention those who think of AB " more of B 
attained by motion from A than of the quantity of length in AB." 
I then refer your Paper on pure time to this mode of thought, 

* A mistake. It was read June 22, 1835. 
VOL. III. X 



306 Correspondence between Sir IV. Rowan Hamilton and 

from which I suppose I could then have pointed out similar modes 
of thought in others ; but I do not remember them now ; I suspect 
the pure time swallowed them up. 

* This notion becomes very conspicuous in the first sheets of the 
book now in hand, in the definition oiA-B, as distinguished from 
OA - OB. Quere, might not (A-B}0 be written for the secood ; 
and in (A-B}0 = AO-BOA B being points, and AO BO 
lengths is there not a capacity of interpretation ? 

'The special subject-matter of quaternions certainly owes 
something to the now established theory of *>/- 1, of which I take 
it to be the legitimate extension. It does not follow that the 
theory of */- 1 suggested it to you, because the interpretations in 
the Paper on pure time may have done it. On this you know 
best. 

' One very leading feature of the quaternions, and one which 
takes the discovery out of the class of extensions commonly so called, 
is the non-permanence of all the rules of algebra as of AB = BA. 
I say takes the discovery not the thing discovered but the process 
of discovering. 

* For one, I long contemplated something like 



gvng, per se, 



but I did not dream of ij differing from ji. 

' 1 must now stop ; but I shall go on by next post. I want to 
look at some old letters of yours before I say anything more.' 



From the SAME to the SAME. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
1 December 31, 1851. 

* I hope you received my letter. I had intended to follow it 
up immediately, but the affairs of a relative intervened and took 
me off all work. These, however, are settled, and I hope to sit 
down to write further to-morrow.' 



Professor Augustus De Morgan. 307 



From SIR W. R. HAMILTON to A. DE MORGAN. 

1 OusteRVATOBY, January 3, 1852. 

* I did receive your letter, and thought it only too favourable to 
myself. As to </- 1, it had haunted me long, and I did know the 
outlines of double algebra, not only before I thought of the 
quaternions, but also (I think) before I had formed, partly with 
the help of Kant, any very definite views about pure time. Still, 
you conceive, that when those views, rightly or wrongly, were 
formed, I was naturally led to see that any number of independent 
progressions might be imagined, as easily as two, and thus formed 
early that notion of "triplets and sets, of moments, steps, and 
numbers," which you refer to, as mentioned near the end of my 
time Paper. (I quote just now from memory.) At least as long 
ago as 1834, I formed plans for several different triplet systems, 
and even contrived several geometrical constructions, but was not 
satisfied with any of them. If you have preserved a letter of mine, 
which I wrote on receiving your first Paper on the foundation of 
algebra or on some connected occasion it is my impression, but 
r don't venture to assert it, that you will find some statement to 
she effect, that if my views about time were correct (and in some 
degree they have since been modified, so as to admit more of the 
aymboKc element), there must be triplets, and even "polyplets." Cer- 
tainly this last word was in my head, but I can't say whether I 
ventured on writing it to you. After a while, I may recover 
among my manuscripts memoranda of some of those old specula- 
tions of mine about sets. I know that I did not assume, in them, 
the commutative principle of multiplication as necessary, although I 
aimed at it. And although I was induced to print (in perhaps 
the Supplement to the December number of the Philosophical 
Magazine, for 1844) a hasty letter of mine to John Graves, written 
in October, 1843, just after the ijk were found, and professing to 
give an account of the process through which I had gone on the 
preceding day, yet I have memoranda which show that I had been 
recently reading my own Paper on algebra, and seeking to illustrate 
its first principles to my boys, especially as related to equal and to 
successive steps in time. So that although, at the last moment, it 
was geometry that moulded and fixed my conception of the y'k, I had 
been prepared for accepting it, by recent as well as by other specu- 

x 2 



308 Correspondence between Sir W. Rowan Hamilton and 

lation, of a more abstract sort. By the way, you have perhaps 
never heard that the part of the postscript to my letter of October 
17, 1843, which was afterwards suppressed in the printing, was a 
permission, " you may show," not a commission, " please to show," 
"this letter to De Morgan." You are most welcome to regard 
this fact, which I can prove (although I do not blame John Graves 
for using his discretion as he did, perhaps fearing to pester you at 
the time), as evidence, or indication, that your writings had in- 
fluenced my thoughts, although, at this moment, I do not call to 
mind any one distinct suggestion which I had not foreseen. 
Perhaps, when I read again the early Papers on the Foundation 
of Algehra, I may come to remember something.' 

From A. DE MORGAN to SIR WILLIAM HAMILTON. 

' 7, CAMDEN -STREET, CAMDF.N TOWN, 
1 January 4, 1852. 

* At last I get an evening to resume my talk upon quaternions. 
I had got so far, I think, as to tell you all I knew of the possi- 
bility of any one supposing you were directly indebted to the 
algebraists up to 1833. Since I wrote I have been trying to 
master the quaternions as you now present them, and I find it no 
child's play. I do not think I can venture any criticism upon the 
whole matter : one or two points on which I thought to raise some- 
thing have broken down on further reading. 

4 With regard to your main question, however, I am clear that, 
so far as you have gone beyond double algebra, whether in the 
way of extension or of new introduction, you need not fear having 
anything which you ought to acknowledge. You are, I have no 
doubt, under general obligations to some of those who thought 
about first principles in the period anterior to 1844 ; but it is in 
that general way in which you are indebted to all the authors of 
your early reading. The exhaustion of the difficulties of a + b^/- 1 
suggested to you, in the Paper of pure time, the idea of "triplets." 
All that is peculiar to quaternions is, I have no doubt, beyond the 
reach of anyone except yourself to claim. It is not yet time to 
decide upon the relation of quaternions to triple algebra : from 
your earlier publications I thought the connexion more visible 
than I am now clear about. 



Professor Augustus DC Morgan. 309 

' As to double algebra, there is a double algebra in the Pure 
Time, which, as far as ideas and leading thoughts go, may very 
likely put you beyond much obligation even to that. 

1 Were I you, I should give in the preface the best account I 
could of the history of my own mind on the subject ; and should 
invite any persons who could furnish you with coincidences of 
thought to forward their references. I feel satisfied you will be 
let quite alone. 

' This is a short summary of a good deal of inquiry ; but you 
know that the plea of nil debet, when the only one, would defy the 
most verbose draughtsman to make it very long. 

* I have looked at the Paper of Grauss, which Wheatstone says 
the Germans, with one voice, cite as turning their attention to the 
subject. Nothing is there which can in any way suggest anything 
analogous to quaternions or even to double algebra. 

' I am acquainted with Ohm's work on algebra, translated by 
Ellis ; and I am sure you are safe from him. A man who founds 
algebra on the seven independent operations of addition, subtraction, 
multiplication, division, involution, evolution, and taking of loga- 
rithms, is yet far from the root. I have alluded to this in my 
Double Algebra, p. 166. 

' You referred to your use of the term scalar. I hold this and 
others to be moveable terms ; modulus is another ; co-efficient ought 
to be one. 

' Now as to what is of more consequence than the priority 
question, owing to the manner in which I have no doubt that 
question will drop dead. I cannot too strongly press upon you to 
do something in the way of bare enumeration of significations, 
for the use of those who. have been thinking over the general 
ground of symbols. Your multiplied explanations, essential as 
they may be to learners, will drive the initiated to absolute 
despair. 

'A dozen pages of bare definition, with the curb held hard 
upon the illustrative power, be the difficulty to you what it may, 
will put the proficient algebraist in a disposition to read your 
developed instructions with the sense of profit. If you do not do 
this, he will be looking among the things he is up to at a glance 
for the points which he is to stop and think upon the most un- 
satisfactory task I know. But, with a prefix or appendix of sum- 



3 1 o Correspondence between Sir W. Rowan Hamilton and 



niary, he will be let into the pith of the matter without suspense 
or the chance of missing essentials in turning over pages of com- 
paratively matter of course. 

' I was frequently at a loss in what I wanted in the earlier 
pages for want of a complete exposition of a + bi + cj+dk, which 
comes late ; and even now I am in want of a tableau of the whole 
system. 

4 References will he wanted to what you have done in R. I. A., 
in Philosophical Magazine, in Cambridge and Dublin Journal. 

1 1 have not fairly finished the proofs yet, but I am made up 
upon the priority point, and shall not touch it again unless you 
have something to ask. 

' 1 remember a remark of yours, that the want of impossible 
quantities is a defect in double algebra. But the defect is the 
defect of symbols. All those which result from algebra or arithmetic 
are fully explained. But instead of weeping for more worlds to 
conquer, we make them. You surely do not intend any imaginary 
quaternions to remain for ever. However here is the end of the 
sheet ; so, for the present, I await some more print from you ; and, 
wishing you a happy leap year, am,' &c. 

From SIR W. R. HAMILTON to A. DE MORGAN. 

1 OBSERVATORY, January 5, 1852. 

' The note on the other half-sheet, which I wish you to return, 
will show that, besides Christmas or New Year Trees, I, like your- 
self, have had some other business than the quaternions lately on 
hand. But I accomplished my purpose of signing for press the 
30th sheet of my book before the old year was out. 

' 1 never heard of the book you cite, but shall be very glad to 
consult it. Please forward with that view the enclosed line to 
Barthes and Lowell. 

* Yours, in some haste. 

* All compliments of the season/ 

From the SAME to the SAME. 

1 OBSERVATORY, January 6, 1852. 

' I must say a few words more about your letter of December 
li, and my own notices of it. The letter arrived, along with 



Professor Augustus De Morgan. 3 1 1 

several others (from other correspondents or men of business), just 
when I was obliged, or at least wished much, to wind up some 
of business before the end of 1851 ; one of them being 
a series of answers to the Queen's Commissioners, with respect to 
my Koyal Astronomership of Ireland, including a brief report of 
the general nature of considerably more than forty thousand 
observations made here since my appointment ; another business 
being the completion of the 30th sheet of my Lectures. I do not 
know whether you will let me count it under the same head of 
liiNi'iiess at least, though pleasure, it assisted to distract me that 
I was to bring my three children, on the last night of the year, to 
a sort of Christmas festivity at the house of my neighbour and 
namesake, James Hans Hamilton (M. P. for the county), where I 
pulled the little sweatmeat from a tree for your unknown young 
friend, and presented my own little daughter to Lady Clarendon. 
' I must say that I enjoyed the evening at Abbotstown ; although 
/ was very well inclined to come away at midnight, till over-per- 
suaded by your grave-looking acquaintance, my eldest boy, William 
Edwin, who thought we might afford another hour or two, in which 
opinion I could see that my little Helen joined, and even an inter- 
mediate brother, Archibald Henry. But what is all this rigmarole 
about ? Why, simply that I rather skimmed than read your letter, 
at the time of my receiving it, and merely remembered its candid, 
and, indeed, generous tone about the pure time, &c., when I wrote 
my first acknowledgment of it. As to the reference to Gauss, I 
must presume that in my haste I did not even see it at first. Still 
my eye may have just fallen on the name, and then passed on, 
from a scarcely expressed feeling at the moment, that I was quite 
safe there. In fact, with all my very high admiration, before now 
publicly expressed, for Gauss, I have some private reasons for 
believing, I might say knowing, that he did not anticipate the 
quaternions. In fact, if I don't forget the year, I met a particu- 
lar friend, and (as I was told) pupil of Gauss, Baron von Walter- 
hausen (about whom I have a pleasant and not ill-natured anecdote 
to tell you), at the second Cambridge Meeting of the British 
Association in 1845, just after Herschel had spoken of my qua- 
ternions and your triple algebra, in his speech from the throne. 
The said Baron soon afterwards called on me here, and I gave him 
some printed papers, about which he said (or rather wrote) to me that 



312 Correspondence between Sir W. Rowan Hamilton and 

he intended to study them when he should have a little " peace." 
Well, he informed me that his friend and (in one sense) master, 
Gauss, had long wished to frame a sort of triple algebra ; but that 
his notion had been, that the third dimension of space was to be 
symbolically denoted by some new transcendental, as imaginary^ 
with respect to \/-l, as that was with respect to 1. Now you 
see, as I saw then, that this was in fundamental contradiction to my 
plan of treating all dimensions of space with absolute impartiality 
[Tros Tyriusque], no one more real than another. Consequently I 
have ever since held it as a certain and established matter offact 9 
that the great Gauss and myself have been on totally different 
tacks, as regards this sort of geometry. And I have very little 
studied the theory of numbers so little, that although I remarked 
to Herschel at Collingwood, in 1846, that Gauss, in his early work, 
brings in */- 1, I have not even a copy of that former work, and 
had never so much as heard of the later one, till,' on a reperusal 
of your letter, I saw Wheatstone's reference to it. I am very 
curious to know what those biquadratic residues are ; but feel very 
sure that they are not the quaternions.' 

From the SAME to the SAME. 

' OBSERVATORY, DUBLIN, January 7, 1852. 

' Many thanks for your important letter, received this morning. 
It shall be attended to; but I was up all last night for the 



From A. DE MORGAN to SIR W. E. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEN TOWN, 
* Jan nary 7, 1852. 

* Your two received. As to the sugar-plum, small chance of 
its ever finding its way into a child's mouth, for my people voted 
it should be kept to prove to what great mathematicians could 
descend ; it being their fixed idea in spite of my having played 
at blindman's buff within these ten days that such cattlo are 
always thinking of their altitudes, and obscurities, and prolix- 
ities. 

'As to the books, I do not mean to execute your conimi 
until I have tried another plan. It is all very fine to send a 



Professor Augustus De Morgan. 3 1 3 

foreign bookseller to get the books dead or alive that is second- 
hand or firsthand and to give him a hint that you think it 
possible they may cost 10, a hint which his conscience may 
easy in halving, when an untempted bookseller could not do more 
than quarter it ; but it is very clear you do not contemplate form- 
ing a large library. 

' 1 am going down to Maynard's to-day, the little secondhand 
mathematical bookseller, in Earl's Court, Leicester-square, who 
may have both books if so, good. If not, my friend Galloway's 

conies on in February (of which you shall have a catalogue 
and it is well worth looking at), where there is the French trans- 
lation of Gauss's Disquisitions Arithemeticae, and a volume con- 
taining ten of his tracts, which very likely has the one you want. 

4 You know of course that J. Graves has become a great book 
collector. 

* You will have more of me in a day or two. 

' January 8. I went to Maynard's no success. I have written 
to Sotheby to send you a catalogue of Galloway's sale. I think 
you had better wait for it, for it is a mere chance that Lowell & 
Co. can furnish Gauss's Disqumtioncs Arithmeticae, which is quite 
out of print and scarce ; probably you will get it for about a pound 
at the sale. 

* Your Christmas party may count as business amusement is 
very hard work, when it has to be done in a set time and manner. 
Your statement of the fact that your children made you stay 
longer than you intended is one I could confirm by the like. 
Shall we publish the joint discovery that there is no getting young 
people away from Christmas parties ? Though you communicated 
it first, I can prove that I knew it. But I could publish a more 
wonderful thing which you could not have guessed. At a little 
party at my house, on Christmas Eve, was a pretty young French 
girl of eighteen, the daughter of people of condition brought up in 
France (Paris) not devotees, nor the least inclined that way of 
wealthy connexions and wealthy herself well educated wlio /< 
hud seen two or more people dance together, anywhere until she saw 
the i/oiuuj people dance on the evening in question at my house. 

' Your sheet of abstract arrived with the letter. I see you find 
that Walker thinks. 

* I will send you a short account of Gauss in a few days. I 
mean of the quadratic residues/ 



314 Correspondence between Sir W. Rowan Hamilton and 



From SIR W. R. HAMILTON to A. DE MORGAN. 

4 OBSERVATORY, January 8, 1852. 

* Your letter of the 4th, which only reached me yesterday, was 
very satisfactory on the point of priority. I could only acknow- 
ledge its arrival, and even now can scarcely do more than thank 
you for the advice which it contains, and of which I shall not lose 
sight. It is clear that I must write some condensed summary of 
the contents of so diffuse a volume, or it will be thrown aside 
unread by the busy and advanced, even if it should not deter mere 
learners, although the liberality of the heads of this university will 
enable me to offer the book at a moderate price, not yet fixed, but 
to be low. You cannot think it flattery, if I say that its style was 
not designed for readers of your calibre, although I shall be very 
glad if you can find time and patience to read it notwithstanding. 
In fact you may observe that although alluding often to double 
algebra, I do not assume any knowledge of it, and indeed assume 
usually very little knowledge of mathematics, at least in the earlier 
and larger part of the volume, so far as hitherto printed, although 
I am getting just now a little deeper into ellipsoids, and shall give 
a touch of differentials, integrals, variations, &c., in the eighth and 
closing lecture, which I have nearly reached. The first was put in 
type so long ago as 1848, and a great part of it was designed to 
be, nearly word for word, what had been actually delivered in my 
lecture room in the June of that year, Mr. Salmon of Dublin, and 
Mr. Cayley of Cambridge, doing me that day the honour of attend- 
ing. Distractions occurred ; and, when I resumed the task of composi- 
lion, it seemed convenient to continue something of the lecture style, 
although not binding myself to a mere report of spoken matter. 
I trust that ordinary readers will not lose by this, though I 0:111 
well conceive your finding much of the book tedious. It may be 
fair to myself to mention, that while no one but myself is answer- 
able for any detail, both Graves at starting, and Herschel then 
and since, urged me not to spare explanation, and not to i' ;ir 
(liil'useness. Still I own that I have been too diffuse, but it iimy 
not be yet too late, especially as money is not here a difficulty, to 
attach not merely a copious table of contents, in which I have 
already made some progress, but also a concise and readable preface, 



Professor Augustus De Morgan. 3 1 5 

or (so to speak) preliminary essay, into which something of history 
may without impropriety enter. 

* As to you, perhaps you may find it enough to read the seventh 
and eighth lectures (if time and patience allow), merely turning 
back occasionally to some former articles as referred to. I hope 
shortly to write again. 

* Can you assist me to procure Mourey, Paris, 1828 ? 

From the SAME to the SAME. 

' OBSERVATORY, January 10, 1852. 

1 1 did not intend to write to you so very soon again, for I 
don't usually ocenohelm my correspondents at such a rate as I have 
lately been doing in your case ; but just as a messenger was ordered 
round, to take to the Chief Secretary's Lodge, in the Phoenix Park, 
a note from Lady Hamilton, declining in her name and my own an 
invitation to a ball next week one can't always keep up that sort 
of ball arrived your pleasant letter, with the account of the young 
French girl, who had never even seen people dance ; which, by the 
way, is all that I have done since I was very young indeed. Still, I 
do like to see a dance now and then, especially when children enter ; 
and you must blame yourself if your last letter shall provoke me 
to send you soon a copy of an old sonnet, suggested by the sight 
of a boy asleep after a viceregal ball, chiefly for children, in Lord 
Anglesey's time, at least twenty years ago, and recalled to my 
recollection by the little festivity at Abbotstown, in this neighbour- 
hood, lately. Meanwhile I remain, with all compliments of the 
season, very sincerely yours,' &c. 

1 About the books ; the theory of numbers is not much in my 
way, as yet at least ; and I shall be quite content for the moment 
to get even such a general notion of Gauss's quadratic residues 
or biquadratic, if they be so as your leisure may allow you to 
give. So I quite approve of your not sending my note to Barthes 
and Lowell, but shall probably buy a few of Gralloway's books. 

1 Did you look at my solution by quaternions of the well-known 
topical problem, first solved by Apollonius, as I find that Eutocius 
records, and used by Wm. Thomson for electricity ? (See Lecture 
vii., article 459.) It seems well fitted to exemplify some processes 
of calculation. 



3 1 6 Correspondence between Sir W. Rowan Hamilton and 

' I confess myself curious to know to what extent you have 
examined into the working of quaternions as a calculus, as distin- 
guished from their principles as a speculation.' 

From A. DE MORGAN to SIR W. R. HAMILTON. 

' 7, CAMDEN-STREET, CA.MDEN TOWN, 
'January 10, 1852. 

' I can lend you Mourey ; as you ought to see, with this letter. 
You will take care of it I know ; and you may return it at leisure. 

' Mourey would have been very remarkable if Warren had not 
appeared in the same year. 

' I do not think you too diffuse ; on the contrary, you are quite 
right to explain at great length ; but I want a synopsis for those 
who are already in the subject. 

'By and bye, when the French tardily begin to cultivate 
algebra as a science, they will declare that Mourey did it all. So 
I would not on any account lose Mourey/ 

From SIR "W. E. HAMILTON to A. DE MORGAN. 

4 OBSERVATORY, January 13, 1852. 

' Thanks for your promise to lend me Mourey, of which I shall 
take every care when it arrives ; and the post to this place appears 
to be safe, though slow. I heard of it a good while ago, and shall 
be very glad to see it, though I fancy the book to be little else 
than double algebra. If the French want an anticipation, though 
not a very complimentary one ! of the quaternions [see the word 
"absurdes"], I can point out what might with some plausibility 
be claimed by them as such, and what I think will startle you, as 
it did me, when I met it (a few years ago, but long after my own 
views and notations were formed and published), on the occasion 
of my seeking to verify some of Peacock's references to theAnnale* 
de MatheHt'iti'jinx, which I consulted in the Library of T. C. D. 

'Did not Servois show, as usual, great sagacity? At the timo 
I refer to, I spent several hours, on a few successive days (at somo 
inconvenience to myself, not worth explaining), in examining 
what was done by Argand and Fran9ais, near the f beginning 
of the century. Argand (if I remember) was admitted to ha\v 
priority of Fran9ais, and it was claimed that he had published as 



Professor Augustus De Morgan . 317 

early as, or at least independently of Buee, whom I never could 
comprehend, though he has some happy hits, such as " adjective." 
Cauchy adopted this from Buee, and seems to have been long 
familiar (from others) with the notion of -Double Algebra. 

" Copy of my rude Memoranda. 

" SERVOIS. 
" [Page 235, 4th tome ? I forget, and cannot at present verify.] 

" LAFEEE, November 23, 1813. 
" . . . trinomial form . . . 

(p cos a + q cos ]3 + r cos y) 
x (p f cos a + q cos ]3 4- / COS 7) 
= cos 2 a + cos 2 ]3 + cos 2 7 = 1. 

Les valeurs de pqrp'q'r' qui satisf eraient a cette condition seraient 
abxurdes : mais seraient elles imaginaires ? [c. a. d. reducibles] a 
la forme generale, A + B */- 1 ? Voila une question d'analyse fort 
singuliere, que je soumets a vos lumieres." 

* Nothing seems to have come out of this hint . . .My impres- 
sion is that Servois was dissatisfied with the double algebra of 
Argand and Francais, and threw out (with, I must say, great 
sagacity) the foregoing suggestion, to show what other sort of 
analysis would be required for geometry of three dimensions. 
You see that / solve his problem by 

p = i, q=j, r = k,p'=-i, ?' = -/, /=-&. 

1 OBSERVATORY, January 14, 1852. 

' P. S. The Mourey has arrived, and shall be taken every care 
of. In a sense, I have already read it through, but must re- 
consider the proof of the existence of a root. I see that either you 
or I but I hope it will be you must write, some time or other, a 
history of v/^1-' 



318 Correspondence between Sir W. Rowan Hamilton and 



From A. DE MORGAN to SIR W. R. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEN Towx, 
' January 15, 1852. 

* I have several points to write about ; but am pressed just I 
now. 

* I do not hold 

(p cos a -f q cos )3 + r cos 7) 

x (// cos a + #'cos /3 + / cos 7) 

= COS 2 a + COS 2 j3 + COS 2 7 

to be utterly out of algebra. If you attach a condition namely, 
that //, &c., shall be finite it is so ; but so is 

aj-a+1, 

and x = 2x. 

I suspect something of this kind : 

p cos a +p . v/- 1 cos J3 + r cos 7, 

1 1 /-, 1 

- cos a v/- 1 cos p + - cos 7, 
p p^ r 

where p and r are zeros related zeros.. 

* The whole of this subject wants considering. To wit, the 
case of 

A = B = C, 

where A is not = (7, because the conditions of the problem impose the 
form - upon B. 

1 Mourey is nothing else but a part of double algebra.' 



From the SAME to the SAME. 

4 7, CAMDDN-STEEET, CAMDEN TOWN, 
* January 18, 1852. 

* All yours received ; and I shall answer piecemeal. 

'The sonnets are very pretty. I will back a baby against a 
quaternion as a sonnet subject. Now for my mechanical, rule and 
compass, criticism take the following : 



Professor Augustus De Morgan . 319 

' On the son of an old friend "in a light of love" not "with "; 
" passed," accent not pleasant ; many a year gone by ? Last four 
lines idea not flashing enough "fnnn ctir/fc^f frit-nil* : " moaning- 
not very clear without looking at the heading ; whereas the sonnet 
should not require a heading, "yet" this rather suggests "-A now, 
+ A to come " than " now, and + A to come." " To thee one day," 
or the like, or "some day."* 

' On a child asleep, &c. Line 1, then a little too conversational. 
If the first line could be recast, thee might be ousted it comes 
again too soon ; delegated a trifle too political, if a substitute for 
the idea could be found : "Hero-viceroy's " might come in, per/taps 
(only perhaps). All the rest perfect, with one change ; transpose 
the last two lines ; the last but one is now a parenthesis ; the story 
not sufficiently well-known to give it the reader at "descending."! 

' Speaking of what is in common algebra yet to be investigated 

as to the functions of ^, &c., I could not remember when I wrote 

before an instance which had struck me ; but I have it now. 

* If there be a problem essentially of the first degree it is this : 
To draw a straight line through the two intersections of two pairs, 

1. ax + by + c = 0. 

2. a'x + b'y + c'=Q. 

3. px + qy + r= 0. 

4. p'x + qy + r = 0. 

To do this we must make 

ax + by + c + m (a'x + b'y + c') = 
identical with 

px + qy + r + n (p'x + q'y + /) = 0, 
or 

a + ma' b + mb' c + me 
p + -up' q + nq' r + nr" 

and the reduced equations are of the second degree. 

* But, choosing which we please for a term of comparison, say, 

b+mb' 
q + nq" 

* [Compare vol. ii. p. 636]. f [Compare vol. i. p. 511]. 




32O Correspondence between Sir W. Rowan Hamilton and 

we have for solutions 

1. The effective solution, giving (5). 

b q 

2. m = --, = -- 

giving 

a + ma' _ U _ c + me 

p + np' 
but not 

p + np r + nr 

' Take another term of comparison, and we get the same shift 
in another form. 

' Your MS. index is nearly what I want as to what it goes 
to, quite, perhaps ; but the quaternions would require more length 
rather perhaps would get it without warning. 

' It will certainly enable anyone to make a first approximation 
to what he might pass over as understood, so as to go in medius 
res at once. 

' As to a history of ^/- 1, it would be no small job, to do it 
well from the Hindoos downwards. 

* I must stop now ; but I have a matter for a whole sheet 
which I cannot do now.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

4 OBSERVATORY, January 21, 1852. 

' I have just copied, from memory, two sonnets to Sir J. F. "W. 
Herschel, written on the occasion of his return from the Cape, 
when a dinner was given to him at the Freemason's Hall, in 
London, which I, who visit the metropolis very rarely, went 
thither on purpose to attend. Herschel appeared to me to per- 
sonify the combination of science and imagination, and you will 
see that my lines are an attempt to express that thought in verse. 

* If, among your many and deep researches, you have made 
psychology, as a sort of branch of natural history, one of them, 
you may feel some little interest in the following problem, which 
has often puzzled myself. 

' Among the persons who know anything about my existence 
and my writings, I suppose that the majority would admit me to 



Professor Augustus De Morgan. 321 

be a mathematician ; while all, or nearly all, would say that I 
could only be regarded as a poet by courtesy. 

* Does it not seem then to contradict one of the very tritest 
sayings about human nature, that I care little, or not at all, about 
criticisms upon my poetry, such as it is, while I own myself to bo 
actually sensitive on the score of my mathematics ? 

' Wordsworth did me the honour to cut up, in a more slashing 
style than yours, some of my early poems. I think that I was 
less flattered than indifferent although I did most highly prize 
the advantage of an intimacy with him. 

' Slash away at my sonnets ; but spare me, if you honestly can, 
a little praise for the quaternions ; or, what will be far better, 
allow me the honour of assisting you to use them as a calculus ; for 
such they certainly are. 9 

From A. DE MORGAN to SIR W. E. HAMILTON. 

1 7, CAMDEN-STREET, CAMDEN Towy, 
January 22, 1852. 

' All yours received, giving proof up to p. 496 ; and also the 
MSS. on geodetic lines, which I must wait a few days before I 
thoroughly read, seeing that I have the Astronomical Society 
annual report to get up. By the way, we never hear a word 
about Dublin Observatory. I have always supposed that the 
Astronomer Eoyal for Ireland was engaged about things of more 
moment, which he could do, while any decent honest man could 
observe. But you speak of masses of observations : could you not 
give us a few descriptive words for me to put into the report above 
mentioned. You are one of our Fellows, I see by the list. If you 
affirm, send to me before the end of the month. 

' N. B. The words are the Council's, though you may furnish 
them. 

'I write now, however, to get a subject finished on which 
I have been thinking for some weeks quoad quaternions and 
priority. I feel as if a dissertation were proceeding from me. 

' The mathematicians at home and abroad are getting into 

a somewhat fidgetty and excitable state about priority. For a 

great many years I have noticed a somewhat augmenting 

tendency to guard themselves against others, or others against 

VOL. in. Y 



322 Correspondence between Sir W. Rowan Hamilton and 

themselves ; a frequency of reclamation , as the French call it, or a 
studied tone of renunciation, when the rights of another were to 
be saved. Some of the fraternity, but not the most prominent, 
carry this a great way ; their luprjica always has e-yw expressed 
before it, instead of understood ; and, if it were not for the Greek, 
I should say they put the word in italics. 

* Now, according to my notion, a person should just do his best 
to tell the truth about himself and others, and leave the rest to 
providence. There is no more use in trying to guard against 
accidents or disputes than in trying to word a letter. Try ever so 
much, and the phrase which produces more correspondence will be 
sure to be one of those which you never thought about. 

* This is the course I recommend to you, who want some advice 
very much just now. I was rather surprised, all through your 
letters, at the unquiet tone which you held about the rights of 
others, and the unsettled feeling about your own. But as this 
was probably matter of temperament, I merely thought about the 
question, as concerned yourself and others, and never about your 
feeling on the subject, except as a thing which was to be all set 
right by sufficient assurance on the immediate cause of it. 

4 But in your letter of the 6th, when I read it over again, I 
was startled by your reference to my " candid and indeed generous 
tone." Candid and generous ! thinks I I hope not. I was writ- 
ing quite judicially ; and if I were either the one or the other, it 
must have been at somebody's expense. But then it occurred to 
me that it might be with reference to myself you were speaking, 
and this seemed so strange that I set myself to look back through 
all your letters, to see if I could confirm this. And sure enough, 
on January 3, I found you telling me that in a postscript sup- 
pressed in printing, you tellJ. Graves, not commissively, " please to 
show," buipermissivefy, " you may show," this letter to Professor 
De Morgan. And further, that though you do not blame Graves 
for his use of his discretion, yet, that I am welcome to regard this 
fact which you can prove as evidence that my writings had 
influenced your thoughts. All this rather puzzled me I had 
never heard of any postscript ; and, if I had, should never have 
regarded your willingness that I should see a certain letter ns 
proving that anything I had written had either been seen or not 
been used or not by you. 



Professor A no n stiis De Morga u. 323 



* Again, as far back as when you sent me the proof of the last 
page of your E. I. A. memoir on quaternions, I find that you 
rallirr deprecate the possibility of being supposed to have done 
me some wrong you hope I shall be witts/ictf with what is then 
sii'l of my triple algebra memoir. 

' From these things and other little ones of the like kind I 
began to see that whether you know it or not / am one of the 
persons about whom you are anxious as to what they may say of 
your acknowledgments, or failure of acknowledgments. And per- 
haps you rather wished to pick out of me what I thought of myself. 
So that, whereas the Scotch Sir W. Hamilton corresponded with me 
(perhaps) to make it certain that he might catch me, if I deserved 
it, in pillaging him, the Irishman of the same name is trying to 
lay a trap for himself. And further, whereas the Scotchman, 
being seized by a fit of illness, did actually work himself up on 
his sick-bed to believe I had pillaged him, and employed his 
returning health in raising hue and cry ; so it is much to be 
feared that the Irishman, if he should have an attack of fever, 
would persuade himself that he has been purloining from me, and 
would only get out of bed to deliver himself up to justice, on 
which I should say that, though the Irishman has a better moral 
and social position than the Scotchman, yet, as far as delusion 

12 

would go, 6 = . 


' Now to the point. There are two kinds of obligation which 
one writer may have to another general and special. The former 
is their joint work. A comes to the writings of B in a state of 
preparation which makes the writings of the latter suggestive, and 
which, in fact, lays the match to a train already prepared. Tho 
ideas which may start up in the mind of A may be such as none 
but B could excite, and yet such as B could excite in no mind but 
that of A. In this case no acknowledgment is or can be due, 
beyond, at the most, an inclusion of B, on the part of A, among 
the writers by whom he has been benefited, if he make general 
acknowledgments. 

* Special obligation is incurred when A takes from B, specifi- 
cally, that which another could take the ipsuin. corpus of his 
thought, process, or work. Of course it is indispensable to acknow- 

' ledge it, 

Y2 



324 Correspondence between Sir W. Rowan Hamilton and 

' One reason why general obligation need not be necessarily 
acknowledged is, that it need not be necessarily known. Nor can 
any person be sure he does know all his obligations. 

* The chances are strong that a person who feels general obli- 
gation, and knows exactly to what, has a special obligation. The 
chances are as strong that if he is wholly ignorant of special 
obligation, he is as ignorant of the precise character of his general 
obligation. But the most probable thing may not be true. 

' Among the morbid feelings on the subject very often may be 
seen the tendency to imagine that others have general obligations. 
A thinks that, though he cannot exactly lay his finger on B, yet j 
B has been at him. 

' And sometimes there is an anxiety about general obligation 
a fear of having incurred it and no very clear view as to the ' 
difference between it and that of the special character. And 
among the ascertained facts relative to this phase of the complaint, j 
I believe it must be set down that being Astronomer Eoyal for ' 
Ireland is not an infallible preservative. 

1 With regard to myself, you owe me nothing special that I 
will be certain of. If, to use your own phrase, anything I have j 
written has influenced your thoughts, there can be no ground for j 
supposing that you can do me any wrong by appropriating the 
results, you not knowing of any specific influence. As to general 
influence, it must have been exerted by hundreds upon any one i 
man. 

* Had the double algebra never come into the world, neither 
would quaternions ; but in what proportions the share of symbolic \ 
algebra has been contributed by Peacock, Warren, Servois, your 
own self (in the Pure Time), my own self, if you like, &c. &c., or 
any others, I do not believe you can][tell, or anyone for you. A 
hundred years hence, some one looking down upon us all may see 
more of that matter than we do. 

* In the meanwhile, everyone owes it to history not to push ; 
formal acknowledgment beyond its due limits, for the sake of 
being safe in courtesy or safe in honour. It is, as far as it goes, 
an untruth told in the history of the mind, to say that a drU has 
been incurred which has not. And those who attend to hi>! 
know that mischief may thereby be done. The kind of exjMvtu- 

which I observe to exist, of being mentioned, makes it dillk-ult 



Professor A it gust us De Morga n. 325 

for investigators to be quite accurate. How many times I have 
seen a person take up a new memoir, run it over, put it down, and 

say, without a blush, "I just wanted to see if he mentioned my 


' See whether out of all this you cannot get rid of all anxiety 
on the matter. If you can, you will just say what you know, and 
be content with that. If there should be any in the world who 
will look for more than they get, depend upon it they are not so 
good judges as yourself ; and, as to general obligation, we must 
wait a century to find a better. 

1 In a few days I hope to read again about the quaternions. 

* My wife admires your sonnets very much, both as to form 
and matter, i. e. babies. If you can make any more, particularly 
on the same subject, pray send me one for her album. 

' I have not forgot the promise about Grauss.' 

From SIR W. R. HAMILTON to A. DE MORGAN. 

'OBSERVATORY, January 26, 1852. 

* Your long and interesting letter of last week reached me by a 
sort of private hand this morning. At this moment, having barely 
had time to read the letter, I can only say that it deserves to be 
considered, replied to, and acted on. Thank Heaven, I consider 
my present Course of Lectures as finished, and time was it that it 
should, for it had of late perhaps been making me nervous and 
fidgetty, notwithstanding the important circumstance that the 
liberality of the Board of T. C. D. has taken it out of the region 
of commercial speculation and I am not very sensitive to criti- 
cism certainly not at all so to yours ; although I might not like 
to be sneered at by an uninformed or uncandid person, where I 
knew that great labour had been expended, and believed that valu- 
able results had been attained. It is the promise of future service 
to science rather than the actual achievement so far, which I value 
in the quaternions ; but I must assert my belief that, even in their 
present infant state, they constitute a new and powerful calculus 
for the solution of geometrical, and, therefore, also of physical 
problems. 

* You will see that I am writing more /////, in the contents or 
summary, on points peculiar to them, or which I suppose to be 



326 Correspondence between Sir W. Rowan Hamilton and 



such, than on other subjects, which yet I thought I could not 
avoid introducing in the Lectures. But I must also have some 
preface, which shall briefly state what seems to be the relation of 
quaternions to some other things, especially to double algebra. 
And it has occurred to me that it might be not amiss, as a prepara- 
tion, nor uninteresting to yourself, that I should write to you on 
the chief points of contrast between quaternions and double 
algebra ; the agreements are obvious on the surface, and it will 
cost me little or no trouble to draw up a short account of them for 
others. 

4 All about differentials, integrals, variations, summations, 
general theories of surfaces and curves, with physical applications, 
&c., I must reserve for a future volume. By the time it is ready, 
I may be a little more up to the history of the whole class of 
speculation. Meanwhile you have, on various points, relieved my 
mind ; and don't take any trouble about Gauss at present, so far 
as I am concerned. Kummer's recent Paper suggests to me, who 
know very little about numbers, that Gauss's biquadratic residues 
may perhaps be of the form 

a + ba + ca* + da 3 , 

when a b c d are integers, and a is an imaginary 5th root of unity ; 
but all this is quite out of my way. I may mention, however, 
that I have heaps of unread Liouvilles, Crelles, and Comptes 
Rendus, so that a reference might now and then be of great use to 
me. After a little while, I can send some more old "baby" 
verses for Mrs. De Morgan's album.' 

From the SAME to the SAMK. 

* OBSERVATORY, Monday night, January 26, ISO'J. 

' Is not this an amusing account of the origin of the phrase 
" Panic " ? It is a pleasure, and an useful one, to go off now and 
then from one's habitual pursuits ; and while hunting, just now, 
among my scattered books and papers, for your essay on triplets, 
to write to you about, my hand fell on a book in partially con- 
tracted Greek, which I opened at this story about Bacchus and 
Pan. Come, said I, De Morgan knows Greek ; this will do as well 
as anything else. So I set to, as a relaxation, to copy the story 



Professor Augustus DeJtf organ. 



327 



for you ; but may well have made some mistake in the accents, 
especially where ^/^contracting ; and even, though I don't suspect 
it, in the sense, here and there, as I have not consulted any version, 
lexicon, or grammar, and never met the passage before. I have, 
meo pcriculo, inserted an iota wbtcripfam or two, which may have 
been hazardous. Never mind duriora passi. I have found some- 
thing more interesting, namely, your first note to me ; but one 
thing at a time. Good-night. 

* (I have not even looked &i the title-page, but your bibliography 
may probably anticipate all that).' 

"EK TON DOAYAINOY I1EPI STPATHrHMATON. 



rjv 6 Day. OUTOC Wpwroc TUIV tvptv, 
, K/oa? era;E e(6y KOL \atov' Taurrj rot apa 
Ktpa<T(J)6pov TOV Flaya S^jutoupyouo-ty. 'AXXa Si) KU\ Trpwroe OVTOQ 
aX vo<j)iq KOI Tt\vTQ. *Hv yap Aiovvcrq Iv 
ol <TKOTTOI fjivpiav X^P a TroXtjUtwp 7rc<ctva 
o A<OWffO* ov jur)v oyf Uav. 'AXXa 
<aparm aXaXa^at fjityiarov. Ol juitv 
ai Trcrpat, KCU ro KotXov rijc VQTT^C %x.ov 
Tc TroXe/zto/c fVETroi^dcv. Ot juty Si) 

r?)y 



<ryjuatv vvKTwp 
^AaXa^av' avr//^?jdav 
7ToXXq7 fj 



flavt 



Ilaiu c 



SAME 



SAME. 



* OBSERVATORY, January 27, 1852, 
' Tuesday night. 

*My dear De Morgan. To save time, may not I as well call 
you so, and you call me dear Hamilton ? I respect your English 
prejudice against that sort of thing, but your note of 1841, which 
has just (last night) turned up, reminds me that we WERE once 
introduced. Of course you will say, we could be only once " intro- 
duced " ; but I can prove the contrary, at least by the expressed 
opinion of a lady to whom I have very long looked up, and to 
whom I used, about a year ago, in perhaps a fit of playful spleen 



* cSSeto-ev would please my eye better. 



328 Correspondence between Sir W. Rowan Hamilton and 

(reciprocated at the moment by her), some such expression about 
"introduction." I speak of Lady (Guy) Campbell, daughter of 
Lord Edward Fitzgerald a " Pamela " (of Mad. Grenlis, not of 
Kichardson) with whom, to my own great moral advantage, I 
was intimate an immense time ago, and who always gave me the 
best advice, and the most friendly counsel not that she was so 
much older than myself; and comforted me much about my 
antenuptial troubles, to which that last line in my sonnet to my 
garden referred. Well, what has all this to say to you ? merely 
that meeting Lady Campbell at a Dublin Musical party, given (I 
think) by Mrs. Charles Graves, and at which I met my old pupil 
the present Earl of Dunraven, who was also a prodigious admirer, 
in the innocent sense, of the lady already mentioned, I asked 
Dunraven, in my momentary flurry, to " introduce " me to Lady 
Campbell, on which she (having overheard me) said, in a pretended 
huff, " Yes, that is just the word, you are to be introduced to me." 
I must own that in point of fact I had not seen her for two or 
three years not since she became a widow. 

' Lady Hamilton and I have since visited Lady Campbell, and 
some of her charming children, who live now more than ten miles 
away from us. 

' Fighting off, for the moment, all thoughts about the quater- 
nions (although some curious thoughts about double algebra passed 
through my brain this morning), I may add that it was merely as 
diversion to myself, and as a little mark of sympathy as a Univer- 
sity man with you perhaps you do not know that I am a Doctor 
of Cambridge ever since 1845 that I copied out, last night, that 
little Greek anecdote, or myth, for your reading. It would be 
absurd to boast of scholarship on the score of understanding that 
story. Easier Greek was perhaps never written. I hoped that my 
two boys were gone to bed, at the time of my copying it, but 
found afterwards that my younger son, Archibald, who is a tre- 
mendous bookworm, was down stairs. So I showed him my copy 
of the Greek, and he translated it off-hand. I begged of him to 
search in Homer whether there was any case of the spelling SSTc. 
He brought me first (this evening) the line 0', 138, beginning : 
Aei<r ' 07* tv Ov^tf ; but afterwards hit off what I wanted, by 
producing the line a', 568 : 

*Qc tyar', tcStiviv & /Socoinc irorvta "H/orj. 



Professor Augustus DC Morgan . 329 



I remember writing dun'oni, instead of yraviora passt. Keep never 
minding. 

1 P. S. (1). The said Archibald and myself read the first six 
books of Euclid through together, a few years ago, in the ( 
not skipping the Fifth Book, although I left it to him to decide 
whether we should do so. We agreed in voting that the last pro- 
position of the Fourth Book, about the Quindecagon, was Ana- 
lytical 

1 P. S. (2). Bishop Butler says, in the preface to his Sermons 
(I quote from WhewelPs edition), " Hence an argument may not 
readily be apprehended, which is different from its being mistaken ; 
and even caution to avoid being mistaken may, in some cases, render 
it to readily apprehended" The italics are mine. 

1 How abominably interlined and corrected this note is ! Yours 
is a much better plan, of making no alterations. They don't even 
add to clearness always, although, as you may imagine, it is for 
that purpose, and not for elegance, that I make them. See the 
quotation from Butler, on the margin of the first page ; and pray 
admire my modesty in making it ! ' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
1 January 28, 1852. 

' What I return of course I received. You are doing exactly 
what I want ; there is nothing for it but to be full very full in 
the statement, the most dogmatic statement, of all points of 
meaning. 

* The sonnets are very pretty. I shall return them ; they seem 
to be originals, or at least not lately copied (I do not speak of the 
sonnets, but only of paper, pen, and ink). 

' You certainly understand those abominable Greek contractions. 
I cannot find any reference to this story of Polysenus in Smith's 
classical dictionary (Art. Pan), though the references are very 
thick, and several instances of the " panic " modi of exciting fear 
are given. I suspect the story has dropped out of reference books. 
The story of Pan's horns is curious. It reminds me of the way in 
which the Hindoo mythologists represent military affairs. In their 
great war (answering to the Trojan war in Greek, from which all 



33O Correspondence between Sir W. Rowan Hamilton and 

the great families claim descent) it is said that the enemy attacked 
in the shape of a serpent with many heads. By comparing this 
description with others, it would seem that all represent modes of 
attack, and that the above* represents that the dense column of 
march formed itself into lighter columns of attack, which is more 
military, no doubt, but not half so poetical. 

'All manner of things prevent me going on to-night.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

* OBSERVATORY, January 31, 1852. 

4 In an early letter of what I suppose I may call the present " cor- 
respondence," you invited me to tell you, in confidence, who it was 
that I suspected of robbing me. On further reflection, my grounds 
of suspicion seem weaker than they at first did perhaps you might 
be able to remove them altogether ; but on the whole, I believe it 
is best to be silent on the subject, lest the recollection that I had 
expressed a doubt of the fairness of the person I allude to, and with 
whom I have never yet come personally into contact or collision r 
might make it somewhat difficult for me to be on cordial terms with 
him hereafter, if we should ever meet, or engage in any sort of 
correspondence, private or public. I might have the feeling of 
having acted unjustly, and be in consequence sore, which at present 
I am not ; nor do I at all imagine myself to be in any danger of 
having the quaternions attributed to him. Very generally, they 
have been received by eminent men in these countries, not only 
with candour, but with generosity you must allow me to use that 
word, because I do not see that men who have important and 
original matter of their own to attend to are bound by justice to 
examine into the merits of every new speculation proposed, nor, 
having examined, to praise it, even if it be held to deserve praise. 
For my part, I should be sorry to think myself obliged to 
every new work of merit, even among those which I may 
to possess and to be capable of reading. Nor is it my busiiu- 
tell the world that I admire even what I do admire, unless inl 
I am conscious of some specific or general obligation, definite enough 
to be relevantly told, in connexion with something of my own, 

rring to an illustration representing a many-headed serpent. 



Professor Augustus DC M<>rym. 33 I 



which something, moreover, I am publishing, or have publish 
Now, Horschel, Cay ley, Donkin, Peacock, yourself, and others in 
England, to say nothing of my Dublin friends, have, as it seems 
to me, stepped out of their own ways to recognise and encourage my 
exertions ; not that they have not all shown themselves abundantly 
capable of working in the same line; and you must permit me to 
consider this as generosity, and as something ornamental in con- 
duct a flesh-and-blood covering of what must still be tin- internal 
framework and support, namely, the principle of justice. In short, 
I don't set up for being an ill-used man. 

'As to my having employed the word to which you objected, 
in reference to a note of yours, I doubtless had a motive ; but till I 
re; id over all your letters again with care I may not remember it 
with sufficient distinctness; nor will it then be worth the troubling 
you with a statement of it. But as I do remember what was in 
my head, when I " hoped you were satisfied " with the concluding 
statement about the history of triplets, annexed to my first series 
of researches on quaternions, in the Transactions R. I. A., I may 
as well mention it briefly. I knew that you had preceded my 
friends, the Graveses, in your particular conception of triplets, and 
that I had preceded you in the general conception of sets, and 
was anxious that I should not seem to be greedy of praise for my- 
self, nor to let my old personal regard for others make me unjust 
to the rightful claims of you, who were, even as far as letters had 
gone, at that time a comparative stranger. 

' Then again but this is a delicate point to touch on I had 
been made cautious, perhaps sensitive, by my intercourse with 
poor Mac Cullagh, who was constantly fancying that people were 
plundering his stores, which certainly were worth the robbing. 
This was, no doubt, a sort of premonitory symptom of that in- 
sanity which produced his awful end. He could inspire love, 
and yet it was difficult to live with him ; and I am thankful 
that I escaped, so well as I did, from a quarrel, partly per- 
haps because I do not live in College, nor in Dublin. I i 
that all this must seem a little unkind; but you will under- 
stand me. I was on excellent terms with MacCullagh; was tho 
reporter (of course an admiring one) on his first cominui. 
tions to the Eoyal Irish Academy; spoke of those early pa; 
of his, in 1832, to the British Association, when it first met at 



332 Correspondence between Sir W. Rowan Hamilton and 

Oxford ; took pains to exhibit the merits of one of his papers on 
light, in a (subsequently printed) Address from the Chair of the 
E. I. A., on the occasion of presenting to him a gold medal in 
1838, the first during my Presidency, and for the awarding of 
which to him I had previously spoken and voted in Council, as 
against a paper of my own (a copy of which Address I can send 
you, if you have it not) ; wrote on his melancholy death the sonnet 
herewith enclosed; followed his coffin on foot from the College 
through the streets of Dublin ; co-operated in procuring a pension 
for his sisters ; and subscribed to the Mac Cullagh Testimonial. 

' He was one of the very first persons to whom, in the Council- 
room of our Academy, on the 16th of October, 1843, 1 showed the 
then just born equations involving /,/, k. At that time they seemed 
to him quite new ; but about a year afterwards he worked himself 
into a fever of suspicion, that I had in some way stolen them from 
a " question " of his own, which was, it seems, proposed by him to 
the candidates for Fellowship in Dublin, in 1842, and which 
certainly connected, in a very remarkable way, the ordinary 
v/ri, with an ellipse in space. If I can find the proper Calendar, 
I shall copy the question below.* 

* In this instance, however, as before about the conical refrac- 
tion, he came to acknowledge my originality, and not merely my 
independence or priority, but that he had failed to see the things 
I saw, although it may be supposed that a little farther thought 
might have enabled him to see them. And it is naturally a 
pleasant, or at least a comforting reflection, to me, that Dr. Stokes 
an eminent Dublin physician, who is also an appreciator of genius, 
and had poor Mac Cullagh to spend what was the last evening of 
his life with his (Dr. Stokes's) family and himself, but did not, at 
that time, apprehend any immediate danger informed me, as we 
walked side by side in that funeral procession through Dublin, that 
Mao Cullagh talked for a long time (he said an hour) about the 
quaternions, as a remarkable discovery, which he then attributed 
entirely to me. Still, it is possible that his former suspicions 
(arising, I believe, chiefly from the fretfulness of ill-health) may 
have made me, to this day, a little nervous about being Nu*jnr(- 

' I find that one of the " Stratagems " collected by Polyoenus is 

[Ltctwtt on Quaternion*, p. (43).] 



Professor A it gust its DC A Tor van . 333. 

the well-known order ora petite at Pharsalia, lq UVTU TU TTpoowira. 
So I suppose he may be classed with Lucian, and perhaps lie 
i arc n ted that story about Pan; but I assure you /did not invent 
it, although when a boy I once amused myself by composing 
(what I could not do now) a Lucianic dialogue, respecting the 
Eape of Helen, in which I described her tearful irresolution on the 
Grecian strand, and threw in several pretty incidents, for which 1 
should have found it difficult to assign any other authority than 
that of the relating sea-nymph. Years afterwards, this dialogue 
turning up, I had the malice to send it to my poetical friend 
Aubrey De Yere, without any explanation, and he was delighted 
to receive the new fragment from the wreck of antiquity, from 
which it may be judged that his knowledge of Greek prose was 
not equal to his skill in composing English poetry. 

[Here follows an anecdote printed, supra, vol. ii., p. 34.] 

1 The line from Yirgil that I lately alluded to was of course : 
passi graviora, dabit Deus his quoque finem. 

Query, are the quaternions included among the his quoque? I 
thought I was done with them, but my interest in them has 
suddenly revived. However, I shall try to stick to the table of 
contents.' 

From A. DE MORGAN to SIR W. B. HAMILTON. 

' February 2, 1852. 

' . . . Your account of poor Mac Cullagh's death interests me 
much ; it tends to a notion I have in my head, namely, that your 
race is somewhat susceptible on such points as that of priority. 
This is not an English reflection upon the Irish for I am not 
English. I was born in India so was my father so was my 
grandfather and the three countries quoad any special associa- 
tion of mine are undistinguishable in fact I am a Briton un- 
attached. . . . Dr. Mulcahy of Galway has just sent me a good- 
looking book about the altitudes of modern geometry. I knew I 
had heard the name before ; and I now remember that in tlio 
materials furnished to me for the life I wrote of poor Murphy in 



334 Correspondence between Sir W. Rowan Hamilton and 

the Supplement of the Penny Cydopcedia* it was noted that the 
person who found him out was a Mr. Mulcahy of Cork. Is this the 
same person ? If so, I suppose he must be well beyond fifty 
years old. 

' There is another Irishman in whose writings I am interested 
I mean Mr. Salmon. What aged man is he ? How comes it 
that the geometrical extensions take such root in Ireland ? Yourself, 
O. Graves, Mac Cullagh, Salmon, &c., are all up in them. Hardly 
a soul in England cares about them. The pure differential calcu- 
lus does not seem to interest you (I mean you plural) so much, 
except as something to be applied. 

* Nothing is more common than a suspicion of others copying 
inferentially that is, taking ideas from other persons' cognate 
ideas. Now, in the first place, this is fair it is the way we all must 
do. A question in an examination paper may suggest and if it 
do so, it is right it should. 

* Lord Adare I remember twenty years ago when he was a 
boy. He used to be with Dr. Robinson in town, and I used to 
meet him at Francis Baily's. 

' I shall be glad to have the Address you mention. The 
sonnet about Mac Cullagh is appropriate, and I have no little 
carping to make on the language. 

4 When you have quite done with quaternions, you must write 
a book on the undulatory theory never mind the history of v/~i. 
The play is not played out yet. Mine is, however, for to-night. 

* P. S. The remark about caution against mistakes lessening 
apprehension is wonderfully true.f Have you my "Book of 
Almanacs " ? and my " Arithmetical Books " ? 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, February 9, 1852. 

' If you really are as little accustomed to the modern geometry 
as you would have me believe that you, or at least that the 
"English" are, you will stare at the following "proof" of a 
theorem of Desargues,+ which occurred to me the other day. It 

[A most interesting account of Murphy's early life. K. P. G.] 

t [Su//ra, p. 329., P.8. (2).] 

t [Friend of Descartes, and author of a treatise on Conic Sections.'} 



Professor A ugn si us De Morga ;/ . 335 



brings in <\/^ but seems to me to have no sort of connexion \s ith 
double algebra, nor with quaternions. 

* The theorem is that cited at the commencement of my Ab.^t 
lately sent, namely, that if a transversal cut the four successive 
sides of a quadrilateral in a conic, and the curve itself, in the six 
points -4, -4', B,B> 0, C', those points are in involution. J'.v 
imaginary perspective, substitute for the conic a circle, and for the 
transversal a line at infinity, perpendicular to the bisector of the 
angle between two opposite sides of the quadrilateral. Take for 
unity the perspective from the centre on the line, the foot being 
made origin of abscissae, and the line itself the axis. Then the 
points of intersection with the sides come to be represented by the 
numbers + t, + u 9 -t, -u, t being conjugate, as also u ; and the 
intersections with the circle, by V /^T. But whatever i may be, 
the three pairs t, u, i, are in involution ; .*. &c. ! Perhaps I 
have caricatured the mode of " proof," if such it may be called, 
but I assure you that I should feel no small surprise, if a theorem 
derived in this way should afterwards be shown to be false, by 
processes of a more ordinary kind. 

' Salmon may be called a young man, at least he is much 
younger than I am. 

' I think that there is a greater, or at least a more general, 
aptitude for pure geometry in Ireland than in England. The 
Fellows of T. C. D. are nearly all geometers, and some of them 
are extremely good ones, although the public examinations for 
Fellowship do not turn much upon geometry analytics having, 
I think, in general, a larger share allowed them in the mathematical 
part of the course; which course is very extensive, and perhaps 
too miscellaneous. I was very glad to get Mulcahy's book, and 
immediately set my boys to begin it, which has had the good and 
designed effect of driving them back to Euclid. Mulcahy, before 
he was appointed Professor in one of the Queen's Colleges, had 
long been celebrated in Dublin as a " grinder," or private tutor in 
mathematics, and was known to be well up in all traditional lore 
of cuts which I was never taught; for though I always rarrird 
away whatever premiums, &c., were going, when I was an under- 
graduate, I had no private tutor ; and, on the other hand, v 
obliged by circumstances to become one, although I was ait ri\\ aids 
induced to receive two of the young Pagets here, as guests rather 



336 Correspondence between Sir W. Rowan Hamilton and 

than as pupils, although paid for, during Lord Anglesey's first 
vioeroyalty, and at a later time had Lord Adare as a paying 
inmate, but rejected all other applications. Lord Adare, now 
Dunraven, is a distant connexion of mine fifth cousin by my 
mother's side and a very pleasant person he was to have in one's 
house, besides that he was so fond of astronomy as almost to be a 
second assistant. I particularly stipulated with his father that I 
was not to be expected to teach him the classics, but could not 
resist the impulse to give him some help when I saw him at times 
hammering away at Greek. One day that some blunder in the- 
grammar (which perhaps I might now make myself) provoked me, 
I asked him, " What did you learn at Etj>n ? " " Learn," said 
he ; "I never fancied that I was sent there to learn at all." And 
as he was a most dutiful son, I believe that he spoke the exact 
truth. 

6 1 do not admit that the Irish are peculiarly susceptible or 
rather jealous, at least in matters connected with science, although 
no doubt they are, myself included, an excitable race. Poor 
MacCullagh was an exceptional case, and his fretfulness arose 
out of ill-health. I am not conscious of being on terms of even 
the slightest unkindness with a single Fellow of T. C. D., and 
with several of them I am on a friendly and indeed affectionate 
footing. To be sure, many of them ridiculed me about the qua- 
ternions, and Charles Graves once burst out with the excla- 
mation, in my presence : " It is astonishing what a prejudice- 
exists against the quaternions, and that among people who confess- 
that they know nothing of the subject ! " " Would it not be 
more discouraging," I replied, " if the same prejudice continued 
among those who are acquainted with it ? " 

' February 10, 1852. Mac Cullagh's question (for Fellowship 
here, in 1842) was the following, and I think you will admit it 
was a remarkable one, while yet you will see that the yTT is the 
ordinary one, and that not even double algebra (at least such aa 
Warren, &c., have used) is introduced, though each equation is- 
obviously designed to include two. After my attention was called 
to the subject, about the end of 1844, by learning that Mac Cullagh 
kid a stress upon the result, and on its previous publication, I 
eaaily proved the theorem at this moment I forget how and 
deduced some others similar to it ; but was not able with truth to 



Professor A u gust us De Morga n. 337 

say that I had consciously derived from it even the slightest h< 
or suggestion, as regarded the invention of the quaternions in 
1843. Whenever your great work on-v/Il, from the Hindoos 
downwards, appears and / have no intention of writing such a 
work, although I said something playfully about it this question 
may properly have a niche in it. 

[Hamilton then gives Mac Cullagh's Theorem, and adds, * I 
see, since copying the ahove, a very simple proof of the theorem/ 
which proof he accordingly appends. Both theorem and proof are 
to he found at p. (43) of the Preface to the Lectures on Quaternion*.'] 

1 Salmon is an excellent fellow. Mulcahy must be as old as 
you conjecture. I have not those books of yours which you 
mention nor your Algebra ; but I can't expect you to give me 
all your books.' 

From A. DE MORGAN to SIR W. R. HAMILTON. 

'7, CAMDEN-STREET, CAMDEN Towx, 
'February 12, 1852. 

'I don't see the relevancy of your appendiculum to your 
answer as to whether you had certain books of mine namely, 
that you can't expect me to give you all my books. "Who said you 
did or could? I trust you don't mean it as a reflection on my 
fecundity of printing. I say this because I did collect all I have 
printed out of serials, and was startled at the mass ; and perhaps 
you may have been startled too at the idea. Nevertheless, I shall 
send you the books I mentioned as soon as the new postal regula- 
tion comes into force about books on March 1. This sending of 
books, you know, is done by authors quite as much for their own 
gakes as for their friends. 

' Your imaginary perspective I can admit. Concealed under it 
is a substitution of the equilateral hyperbola for the circle. An 
assumption of x* - if = 1 for # 2 + if = 1, is, of course, // ^/~\ for y as 
a mere instrument of transformation. I have always held it a great 
pity that the equilateral hyperbola had not some marked property 
like that of the circle which would have made Euclid introduce 
them together. 

4 A year or two ago I wrote a letter to the PMhtopkioeU 
Magazine, signed " The General Equation of the Second LK _ 

VOL. III. Z 



338 Correspondence between Sir W. Rowan Hamilton and 

complaining, in the name of that respectable member of society, 
of geometers having made it appear as if the ellipse took more 
than its share of a certain property. Both in the ellipse and 
hyperbola the locus of the perpendicular on the tangent from the 
focus is the circle an ellipse. The ellipse has the lion's share of 
this property. Now what is the property in which the hyperbola 
takes both ? You will not be long in finding it. 

' I shall draw up a small Paper to show how the modern geome- 
try may be worked without ambiguity on the system of signs which 
I gave in the Cambridge and Dublin Journal. I have been looking 
at Salmon's book lately, and regret much that his general proposi- 
tions are worked on a case of the diagram, their generality being 
secured by trusting in algebra for the other cases. 

' As to people ridiculing quaternions, let them do it ; but do 
not let them succeed in making you feel it. They exist and act 
as Newton said of gravitation. You take care of your " contents," 
and never mind the " non-contents." 

* If Mac Cullagh imagined that the property of the ellipse he 
had assigned had anything to do with any system of v^-i except 
the old one or could suggest anything he was strangely de- 
ceived. The property, so far as appears, has a shade more of 
connexion with the couplets of the Pure Time than with quater- 
nions. In devising the question, he first established # 2 + #' 2 = con- 
stant, &c., and then constructed his equation. 

* I object much to the phrase of six points in involution. 

1 If the ratio compounded of AB to BC, and CD to DA give a 
ratio of equality, we say AC is harmonically divided in B and D. 
Say, AB, CD are harmonics. 

Then if AB to BC, CD to DE, EFto FA, compounded, also 
give a ratio of equality, why not say A B C D E F are harmonics ? 
We have then an harmonic quadruplet and sextuplet, and we 
might have octuple ts, &c.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

4 OBSERVATORY, February 13, 1852. 

'I was at one time well acquainted with Monge's Aml>, 
that be the name of the larger work, lately re-edited by Liouvillc, 
in which he connects partial differential equations with families of 



Professor Augustus De Morgan. 339 

surfaces, and modes of geometrical generation, and also treats of 
involutes and evolutes in space ; but it is so long since I studied 
that work, that I laid aside, in a sort of despair, your Paper " On 
the Connexion of Involute and Evolute in Space," when I received 
it a few weeks ago. But I took it up for a variety to-day, spent 
the whole morning in studying it carefully, and think that I now 
understand it thoroughly. The transformation 



in (21) is very pretty ; I suppose that in getting it you used the value 



for I found it inelegant, at least in my hands, to differentiate one 
alone of the three equations (20). 

4 About the constancy of OPT, the following simple process 
occurred to me. Merely because PP' and XT' are small of the 
first order with respect to PT (without any use of the rujht angle 
TPP f ), while PIT is a straight line, the angle TP'T' is small of 
the second order, and CP'T may be substituted for CFT'. The 
right angle TPP' next gives (as in your Paper) TP = TP' in 
length. Thus the fundamental equation of plane trigonometry, 

CT* = CP 2 + PT 2 -2CP . PT. cos CPT 

is to be differentiated as if CT and PT were constant, and it then 

gives 

CQ . dCP =CP.PT. dcos CPT, 

when Q is the foot of the perpendicular from T on CP. Simple 
as this last formula is, I got it (just now) as the interpretation of 
an expression for the differential of the scalar of the versor of the 
quaternion PT -f CP. I know that you will not, and ought not, 
to turn aside at present from your own pursuits to examine any 
such calculations of mine; but as, at some future time, you may be 
induced to acquire a practical familiarity with their working, I 
shall here record (if you preserve this letter) a few equations 
respecting involutes and evolutes which, after reading your Paper, 
it gave me little trouble to form.* 

* [These equations are to be found in the Elements of Quaternion*, pp. 
and 621, &c., with an acknowledgment of JDe Morgan's share iii the invet,t 
tiou.] 

Z 2 



340 Correspondence between Sir W. Rowan Hamilton and 

' I hope that you will be able to look over my " Contents," 
when a proof-sheet of them is ready from the printers. I am 
taking pains to mark off distinctly what is common to other systems, 
and, in short, to assist a reader who is not a mere learner to skip. 
As intermediate between the frightful length, but paucity, of the 
" Lectures," and the shortness, but multiplicity, of the " Articles," 
I am introducing into the Contents a division into " Sections," 
i., &o., of which there are now xxx. written out, representing 153 
pages of the book. I am aiming to make the Table of Contents a 
readable abridgment of the work ; but must make a few prefatory 
remarks besides. 

' The night before last I attended a grand ball at the Castle,. 
and contrived to pick up a little botany and embryology from 
Allman, geology from Jukes, and news about a poetical friend 
from Anster, besides feeling a fair enjoyment of the spectacle, and 
renewing or forming an acquaintance with two men, officers in the 
army and navy, whom I had not seen since they were children. 

* I met also a niece of Francis Baily, at present Lady Kane, 
and had a talk with her husband, Sir Robert Kane, about the 
mathematical professorships in the Queen's Colleges.' 

From the SAME to the SAME. 

1 OBSERVATORY, February 14, 1852. 

* As to Euclid and the equilateral hyperbola, one would think 
it almost as natural to have considered the case of a triangle on a 
given base, where the difference, as where the sum, of the angles at 
that base was a right angle. Now Thales is reported (may we 
believe those stories ? I hope we may at least believe that Pytha- 
goras discovered the property oi the hypotenuse whatever becomes 
of the story of the hecatomb. Thales, I say, is reported), if I 
remember rightly what I once read somewhere, to have discovered 
thut the angle in a semicircle is a right angle; and no doubt lie 
kiu'w, if so, the more obvious equality of the sum of the three 
angles of a triangle to two right angles. The complementary 

relation A + B -, for the triangle in the semicircle, or in the oirc !<>, 
is therefore probably older than Euclid ; and it seems that the analo- 
gous relation A - B = - might or even ought to have very early 



Professor Augustus De Morgan. 34 1 

<mrred; but this gives the equilateral hyperbola as locus of the 
vertex. I suppose the property is very well known ; it served me 
several years ago to form a neat equation by quaternions for the 
equilateral hyperboloid of two sheets of which perhaps more anon. 
Another close connexion between circle and equilateral hyperbola 
is, that in each the diameter and tangent make complementary 
augles with the ordinate. 

1 Instead of racking my brains to re-invent your property of the 
hyperbola (though I did make a guess or two), I hunted among my 
numbers (unbound, I regret to say) of the Philosophical Magazine, 
for some years past, and at last lit on the amusing and instructive 
Paper by your friend, " The General Equation of the Second Degree," 
No. 225 of third series, page 546 . . . which I had barely glanced 
at when it came out, and did not then know it to be from your 
hand. It gave me no trouble to see proofs of the things stated in 
No. 1 : real perspective gives at once from the circle the property 
of the enlarged ellipse ; but the analogies were new and striking to 
me. The results of No. 2 seemed more surprising ; I was content 
to prove them by co-ordinates, with the use of which I was once 
expert. Not at the first look seeing that you designed the ellipse 
and hyperbola to have a common minor axis, in one received sense 
of that phrase, I eliminated a?', y' ', between the three equations 



y f (x - e) = hx'y, and found [_(x - e) 2 #'~ 2 =] 



which gave { (x - e)* -<y*}(x*-y z -1)= (I - e* - 
You designed 1 - e 2 = A, which gives x* - if = 1. 
' (To suppose x-e = y would give y' = />./, 

#"= = e -\ .'. (say) x = + e~\ y= + fur 1 , x-y = e; 

the tangent and the supplemental line seem in this case to coincide, 
each making 45 with axis of x. I picture to myself an hyper- 
bola, h > ; but of course h may be < 0. I have not considered 
with any care the meaning of the factor (x - e) z - if. Perhaps you 
have.) 

* I wonder what put the theorem about those supplemental 



342 Correspondence between Sir W. Rowan Hamilton and 

lines into your head. You have explained what led you to wish to 
discover some such theorem. I am very far from knowing all that 
is (perhaps even quite commonly) known about the conic sections ; 
I ut it is my impression that much still remains to be discovered 
about them. 

* Now for a bit of " dogmatism." It is my will and pleasure 
to believe, and you need not try to argue me out of it, that a circle 
is not a degenerate ellipse, but an ellipse a deformed circle. The 
( irde is (I hold) naturally simpler than the ellipse subjectively of 
course but no one need seek to persuade me that it is an accident 
of education our thinking so. It has a "prerogative " as has the 
sphere in space. But I am willing to grant that the equilateral 
hyperbola, and both the equilateral hyperboloids, are also preroga- 
tive figures, very nearly, perhaps quite, as much so as the circle and 
sphere, and far above ellipse and ellipsoid. Ditto about the equi- 
lateral or right-angled right cone. Accordingly quaternions give 
charmingly simple expressions for these various eminent loci, as 
possibly I may find time and room to print, in what I hope will be 
my closing sheet, 2 L ; although very decently concise equations 
have long since offered themselves to me for other curves and 
surfaces of the second order ; and for some others, for example, the 
surface of elasticity, and Fresnel's Wave, when treated by my 
calculus.' 

From A. DE MORGAN to SIR W. R. HAMILTON. 



* 7, CAMDEN-STREET, CAMDEN TOWN, 
'February 15, 1852. 



* Don't be dogmatical without any need nobody denies it. 
The circle and the straight line have a real subjective distinctness, 
as the exponents of pure translation and rotation, as well as for 
other reasons. Moreover, though the circle is an ellipse, yet it is 
not an ellipse only. For it would be easy to make a class of 
curves 

s"$0r, n).dx+ \ ^(fc*).<ty-s, 

Jo Jo <5 



such that 0(;r, 1) 1, \//(y, !)!, giving a circle when w = l, and 
never any ellipse, but only a most transcendentissimal unintegrable 
in every other case. 



Professor Augustus /)< M ; ,/>/. 343 

< For like reasons I deny that an ellipse is necessarily a deformed 
circle, though that is one way of viewing it. 

'I believe in projection as a natural and necessary mode of 
deducing the ellipse. But what distinguishes the straight line and 
circle more than anything else, and properly separates them for the 
purpose of elementary geometry? Their self-similarity. Every 
inch off a straight line coincides with every other inch, and off a 
circle with every other off the same circle. 

* Where then did Euclid fail? In not introducing the third 
curve which has the same property the screw. The right line, 
the circle, the screw the representatives of translation, rotation, 
and the two combined ought to have been the instruments of 
geometry. With a screw, we should never have heard of the 
impossibility of trisecting an angle, squaring the circle, &c. It is 
true that the assumption of the screw is very like an assumption 
of all the point of difficulty ; but it is in natnra renon that some 
assumption there must be arithmetic excluded. 

' I admit the prerogative of the right cone, right cylinder, and 
sphere. I doubt about the cone, though, altogether, as not self- 
similar. The right circular cylinder and the sphere are the right 
line and circle of solid space ; they are also the simple translation 
and simple rotation of a circle. 

* So we have 

fsimplest translation right line.f 

* Point 

[simplest rotation circle. 

fsimplest translation plane. 
'UBightline. . 

[simplest rotation cylinder. 

["simplest translation cylinder. 

*Circle 

[simplest rotation sphere. 

* All self -similar. 

' Now because of the double genesis of the cylinder a trans- 
lated circle we get the screw self-similar (a very bad word). 

* But if you want to bring in the cone which, though rotation 
of a line, is translation only of groiciny circle, and, more"-. 
simplest rotation of line I do not see why the not-siniplrst rota- 
tion of circle should not come in ; and then you get the ring and 



344 Correspondence between Sir W. Rowan Hamilton and 

the self-cutting ring with the orange and the large pip in the 
middle. 

4 1 never meant to write so much to night. 

' You do not see how I get my supplementals. I considered 
that there must be a property unfound, and looking at perpendi- 
culars 



--. 

- in two senses reciprocal of a), I conjectured that the simpler 
reciprocal formations 



must play some part. And then I felt about for it how I don't 
know ; but everybody who has ever played at blindman's buff has 
some idea of the process, if ever he has had the handkerchief fairly 
on. 

' 1 have written the inclosed to Mr. Salmon of whom I know 
nothing but his books, though that is a good deal, or a deal of 
good. Read it for the demonstration, and pass it on to him if he 
is a man to like such things from a perfect stranger. Entre -nous, 
since my grande guerre with your Scotch synonyme namesake I 
mean I think twice before I open correspondence with anybody 
unknown to me.' 

4 P. S. The evolute equations I dare say I shall pick up in the 
course of the content-reading. 

' I have known Lady Kane a long time. When Francis Baily 
went over to Dublin to the British Association, she introduced her- 
self to him. He hardly then knew he had such a relation. After 
that she was a favourite of his, and is of many people. 

* I have just got Mr. Salmon's book on plane curves ; he really 
is a valuable writer. It is a great comfort to have the journals all 
sifted and arranged, particularly for persons like me, who never 
strive after the mathematical newspapers, and whose knowledge is 
therefore not always entitled to the motto signatum prcBsenti nota* 



Professor Augustus De Mot 345 



From SIR W. R. HAMILTON to A. DE MORGAN. 

'OBSERVATORY, February 21, IK 

' The proof for Salmon read, understood, and now forwar 
loy me to the College ; but he may not receive it until Monday. 
I have said that you seemed to fear it might be thought an intru- 
sion, and left the forwarding of it to my own discretion an this 
instance ; but / am sure that Mr. Salmon, who is a very amiable 
man as well as a good geometer, will take it, as he ought, in good 
part. I had other things to write about, but must postpone them.' 

From the REV. G. SALMON, F.T.C.D. to SIR W. R. HAMILTON. 

TKIKITT COLLEGE, February 24, 1852. 

1 1 have to thank you for your letter enclosing one from Prof. 
De Morgan. You have judged rightly that I should indeed have 
been sorry had you adopted the course of suppressing the letter. I 
answered his letter yesterday, and will now give you, as you wish 
it, the substance of my remarks on his proof. 

' The following demonstration is the same in principle as De 
Morgan's, but simpler in form : 

[Here follows a proof of the anharmonic theorem.] 

' I then took an opportunity of expending on De Morgan (I 
Relieve now rather unprovokedly) some pent-up wrath against the 
English school of geometers generally. They seem, in general, not 
convinced by the most elegant geometrical proof until it is helped 
out by an algebraical demonstration. For example, Mr. Walton, 
of Cambridge, has lately published a collection of problems on 
conic sections, where he gives himself much credit for replacing 
the geometrical proofs in mine and other works by algebraical 
ones. Now it seems to me that there are real difficulties enough 
in mathematics, without increasing the number by conventional 
restrictions. Why are we to forbid ourselves the use of any in- 
strument which can help us on ? When trying to make out a new 
theorem I would employ, with equal willingness, algebra or 
geometry (or quaternions, if only I could use them well enough) ; 
-and, if any mathematician could invent for me a new method, I 



346 Correspondence between Sir W. Rowan Hamilton and 



should use it with thankfulness. And if this is the method which 
I employ in my own researches, I think I should be wrong in 
teaching my pupils a different system. To employ algebra exclu- 
sively would no doubt make the book prettier and more harmo- 
nious ; but it would train up my young readers in the exclusive 
employment of a single method when it is desirable to make them 
dexterous in the use of all, and to set them an example in using 
all by turns, as most convenient. The case is different when the 
object is not to teach a beginner how to investigate mathematical 
truths, but to perfect new methods. Thus I can understand that 
you would wish to try the powers of the quaternion calculus on all 
] >ossible theorems ; and the less manageable the problem appeared 
l>y quaternion methods, the more anxious you would be to make 
the quaternions overcome the difficulty. But when once it has 
been satisfactorily ascertained that all manner of problems can be 
solved by quaternions, it seems that a writer on geometry gene- 
rally ought only to employ quaternions where the proof by them is 
simpler than by the other methods, geometry where geometry is 
the simplest, and ordinary algebra where it affords the shortest 
path to truth. 

* I have been led into these remarks, because I think the proof 
in my Art. 262 is better than that which De Morgan (from love of 
uniformity and system) proposes to substitute for it. At least you 
will see (Higher Plane Curves, p. 137) how my proof at once gives 
the corresponding properties for higher curves. 

'Can you oblige me with a reference to your proof of the 
anharmonic theorem ? . . . 

' Your book will probably make the use of your method more 
general. At present it is a bow of Ulysses, which no one can bend 
but the owner.' 

From SIR W. R. HAMILTON to REV. GEORGE SALMON, F.T.C.D. 

[FROM A DRAFT.] 

< February 27, 1 

' I think you show your usual good sense in your remark 
the propriety of not being bigoted to any one method in mathe- 
matics. For the very reason you mention, I hope to continue 

lying quaternions, from time to time, to all sorts of geo- 



Professor Augustus DC Jforyui. 



metrical (and also to some physical) problems, but shouM be 
very sorry to forget co-ordinates, or not to learn such abi '. 
aii I powerful notations as you treat of, or to lose the pleasure and 
the profit of reading occasionally in their own noble language the 
writings of the Greek geometers. We may live to see a sort of 
mathematical glossology grow up a comparative anatomy of the 
structures of several distinct systems of expression on geoni* 
subjects. It has amused me to fancy sometimes a demonsti 
of Apollonius printed in one column of a book ; a Cartesian inves- 
tigation of the same theorem in another column ; one on the plan 
you favour in a third ; and a quaternion calculation in a fourth. 

4 1 have been so hard at work in winding up my book, of which 
I have almost finished writing out an elaborate table of contents, 
[though &feic articles remain to be added to the text, that I have 
reserved, not read, your proof of the anharmonic property. But as 
I understand De Morgan's proof, I hope to understand yours also. 
Mine, such as it was, appeared in the Philosophical Magazine for 
October, 1846, and in the K. I. A. Proceedings for July, 1840.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
' February 23, 1852. 

' 1 received the new page of contents as I was examining the 
old one. My examination does not lead to any fault-finding for 
I recover very easily all I distinctly apprehend and no table of 
contents can do more for anybody. 

' One point that I wished very much to gain will be gained by 
the table. Your man who is too old to learn new things, as you 
know, will, in nine cases out of ten, pronounce that the new thing 
is useless, if he can get a chance. If the book had been no wl it-re 
condensed into a body of distinct enunciations, he would have had 
a chance, for he would have given his opinion as of a speculation 
about which nobody could contradict him without reading the 
book. But he will be a bolder man who would pronounce in that 
manner, when anyone who looks at the contents can see a syste- 
matized body of results. Every book should be provided witli 
some palisade against mere talkers. And among the mere talkn-s, 
so far as mathematics are concerned, are to be ranked three out nt 
four of those who apply mathematics to physics, who, wanting a 



348 Correspondence between Sir W. Rowan Hamilton and 

tool only, are very impatient of everything which is not of direct 
aid to the actual methods which are in their hands. You will find 
the table of contents a useful outwork. 

* Perhaps you will wonder what I am talking about. But I 
suspect the Irish to be more given to pure speculation than the 
English. I have not watched the Astronomical and the Royal 
Society twenty years without being able to support the preceding 
thesis. The English savant is a very practical animal, as he calls 
himself. Any kneading of mind for the use of future generations 
lie cannot see the use of. There are a few exceptions in the uni- 
versities and but a few. 

* The materials of the table of contents will do very well.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

1 OBSERVATORY, February 25, 1852. 

4 It is really a great satisfaction to me to receive, as I have just 
done, your general approval of my table of contents so far as 
you have seen it. Had you objected to the plan a fortnight ago, 
and suggested any special improvement, I should have honestly 
set about to act, if I could, on the suggestion ; but it is now too 
late, if I am to co-operate with Charles Graves this year, as I wish 
much to do. A difficulty about the quantity of small mathematical 
type required had almost driven me to despair; but just before 
joining, yesterday, the assembly of doctors and others, Lord 
Clarendon included, who were to dine in Trinity College (Dublin) 
at the grand banquet, given by the Provost and Senior Fellows 
(which really was a splendid affair), I found a minute or two to 
call at the University Press, and arranged that the contents should 
be printed off in half-sheets. 

1 Lady Hamilton is starting to attend church in Dublin, and I 
must seal and sign. 

* The dinner was the usual one given at " Commencements," 
but on an unusual scale.' 

From the SAME to the SAME. 

< OBSERVATORY, March 15, 1852. 

'The three books have reached me, and are very welcome. 
Something for you very soon. Meantime let me just ask is lliat 
capital word "eliminaut" your invention? I have lately 1 
printing. . . " determinants, or as some prefer to call them, elinri- 



Professor Augustus De Morgan. 349 

. ." Of course this cannot be mistaken as claiming any 
merit for v/cw, but ought I to name you, or rather might I do so 

From the SAME to the SAME. 

' OBSERVATORY, April 2, 1852. 

1 1 did so flood you with letters and papers for awhile that I 
should not be surprised if you supposed me to be dead or ill. 
However, such is not the case. I made, to my own surprise, a 
speech on Monday last, in honour of the poet Moore, and attended 
a private concert (with some " dear 500 friends ") at the Castl- 
the evening. It was chiefly sacred music, and was understood to- 
be given for the sake of clergymen and others, who scruple (which 
I do not) the being present at a ball. To me, who am old enough 
to remember when Moore's poetry was thought to have somewhat, 
or indeed a great deal, of a rebellious tone, it was striking, and 
almost amusing, to hear the final " (rod save the Queen " immedi- 
ately preceded by a melody of Moore's, which lamented that the 
emerald gem of the western world had been (onjz centuries ago) set 
in the crown of a stranger. But I had the honour of being invited, 
in the summer of 1849, to meet the Queen and Prince, at Lord 
Clarendon's Viceregal Lodge, in the Phoenix Park, near this place, 
and a brilliant meeting (for Dublin) it was combining, as struck 
me at the time, the attractions of a Musical Soiree, a Conversazione 
(sotto wee), a Court (for there were numerous presentations), and 
a supper. Well, on that occasion, the chief enjoyment, and the 
chief part of even the pomp, consisted in the singing and the 
pianoforte performance of sundry melodies of Moore. So we have 
her Majesty's permission to admire them ; and, seriously, they 
are not in the least likely to produce any rebellion against her/ 

From the SAME to the SAME. 

'OBSERVATORY, April }'>. \^ 

( I owe an apology to Mrs. De Morgan, for not thanking 
sooner, through you, for her little Paper on Mesmerism in 
nexion with the treatment of insanity. Will you now pivsriit my 
thanks, and please to add that I do not forget my promise to send 
some other infantine verses. 

* [' The name " eliminant " was introduced, I think, by Professor De Morgan/ 

Salmon's Hiyher Algebra, fourth edition: Dublin, 1885. A'utes, p. 3-12.] 



350 Correspondence between Sir W. Rowan Hamilton and 

' As you suspect, I know very little about mesmerism, but am 
so far from being a sceptic or scoffer, that I have acted, in some 
degree, the part of a propagandist, and received some good- 
nntured pity, as such, from Miss Edgeworth, and even from 
Herschel. Not that I ever witnessed a single experiment in 
mesmerism. A squarer of the circle said to me, long ago, about 
some three lines which he had made very neatly to meet in one 
point, but about whose co-punctuality I was dubious, " Won't you 
believe your own eyes ? " Not a bit of me. They have been too 
often suborned, or deceived, by Christmas conjurers for me to 
place that sort of faith in them. Even in what is more strictly, 
or more commonly, called science, my faith in conical refraction as 
&fact (somewhat impudently denied by Mr. Moore), rests far more 
on Lloyd's testimony, than on the experiments of which I was an 
eye-witness. But I do pretend to some insight into veracity of 
character, and cannot disbelieve all that I have heard attested about 
mesmerism. So far as catalepsy and somnambulism, my faith has 
stretched ; but, as yet, the marvels of clairvoyance are beyond its 
reach ; although I have read, with great interest, and a profound 
disbelief, Dumas' "Memoires d'un Medecin." Did I ever tell you 
that Lord Clarendon, one day that the late Marchioness of Lans- 
downe was sitting between him and me at dinner, and that I 
happened to be able to supply the names of some French novels 
("Consuelo" and others), which she had forgotten, turned round 
upon, and looking through and through me with his piercing eyes, 
exclaimed, " Ah ! Sir William Hamilton, when you get />// yourself 
in that Observatory you read something else besides Astronomy ! " 
I am so little of a courtier that I have not yet written my name in 
their Excellencies' books, since my being a guest of theirs rather 
more than a fortnight ago. Babbage would have done it next day. 
Yet I enjoy seeing a Court now and then, were it only to value 
the more my retirement.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STKEET, CAMDEN TOAVX, 
'^ri715, 1852. 

' Where are you now in your printing, aud when do you 
oome out ? 

'Get and read, by due snatches, the History of Physical 



Professor A ugnstus De Morgan . 351 



, by Robert Grant, just appeared. I assure you. 
will find it an extraordinary book for a man quite unknown to 
bring out. But Robert Grant has looked for both matin-mat irs 
and history in original sources. I have had a long conversation 
with the author he owes his opportunities to having been dis- 
abled ten years by a fall. He is a man of business with no 
means and the author of the first complete history of the theory 
of gravitation is helping his brothers in a parasol business until he 
can get a clerk's place for himself. 

I Looking up points of history about the differential calculus, I 
came to the following, which is rather amusing : 

' Laplace who stands out for it that Fermat was the inventor 
of the differential calculus explicitly states that he had applied 
his method to the determination of points of contrary flexure and 
to transcendental curves. On looking at Fermat's tract on max- 
ima and minima, I see that at the end are two diagrams ; 1, a 
conchoid with a tangent drawn near the point of contrary flexure ; 
2, a cycloid. 

Fermat applies his method to neither nor does he mention the 
wntrary flexure of the conchoid. I conjecture that Laplace took 
the report of someone who looked at the pictures only, and did not 
read the text which accompanies them, and who took the flexure 
for granted because the tangent comes near the point of contrary 
flexure. Is it possible Laplace had done this himself ? ' 

From SIR W. R. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, April 16, 1852. 

I 1 am growing savage ! Southey, with whom I once slid down 
a part of Skiddaw, and who read to me half a book of his hexa- 
meter poem, the "Vision of Judgment" and very well, I thought, 
he read it told me that he considered a proof-sheet to be one of 
the pleasures of life. I quite agree with him. Judge, then, of my 
wrath, at finding that the postman has again brought me nothing 
from my printers ! Since Saturday last I have been in this state 
of starvation. And what makes it more provoking is, that the 
rascals sent me word that they could afford no Easter holidays ; 
and I have in consequence deprived my little daughter (Helen) 
and myself of the pleasure of a visit to a cousin of mine at Trim, 



352 Correspondence between Sir W. Rowan Hamilton and 

Miss Hamilton, who learned on my lap, when an infant, some- 
lessons in Sanscrit and Persian not to mention Greek, Latin, and 
Hebrew, or French, German, and Italian, and all things proper 
for an infant prodigy ; the truly prodigious point being that the- 
said cousin has grown up a most sensible, quiet, and prudent 
girl, or young woman. 

''April 17, 1852. This morning came a new slip from my 
printers, bringing the work near to the end of the 35th sheet (560 
pages). About eleven pages, at the end, are actually in type, and 
a woodcut representing the focal hyperbola ; but I have found it 
difficult to resist the temptation of throwing in some intermediate 
articles on other subjects. No doubt this last lecture will be a 
terribly long one, and may seem too discursive or digressive ; but 
I claim to have had a plan, and while sensible of heaps of faults 
in style, &c., am approaching fast to the accomplishment of the 
object that I had proposed to myself. 

' Whether I can do more than lend you the enclosed paper, 
containing selections from some of my mathematical examination 
papers in T. C. D., for the last three years, and designed to ex- 
emplify the connexion between Quaternions and Variations, I am 
not yet certain. But I shall try to procure a few copies for friends. 

* As to Sanscrit and Persian, I do not pretend to read them 
now ; but my childish acquaintance with various languages may, 
as I have often since thought, have assisted me in my maturer 
study of mathematical symbols, and even in my attempts to 
enlarge the limits of mathematical expression.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

1 7, CAMDEN- STREET, CAMDEX TOWN, 
1 April 16, 1852. 

* Yours received before mine was sent. Some of these days I 
will send you a simple narration of fact without any theory 
attached on the subject of clairvoyance and you shall put the 
theory of probabilities to work to make a string of coincidences 
of it. 

'As to my eyes or eye I don't believe them (it) much never 
hod much reason. But my means of judging of clairvoyance 
whatever it is were direct, immediate, and personal. 



For letters which ought to have followed that of April 16, 
1852, the reader is requested to turn to p. 623, infra. 



" Life of SIR WILLIAM R. HAMILTON," 
Vol. III., pp. 352, 353. 



Professor A ugustus De Morgan . 353 

* The following good notion of fluxions and of infinites having 
other ratio than equality is from a bishop who died in 1382 
(not 1832) : 

' " In every semicircle, the intension of the breadth [ordinate] 
begins from the utmost degree of velocity, and terminates at the 
utmost degree of tardity in the middle of the arc ; the remission 
begins from the same middle point with the utmost degree of 
tardity, and terminates with the highest degree of velocity. But 
lest anyone should babble about this, I understand utmost velocity 
as in respect of any other which does not appertain to the same 
kind of figure ; for I do not deny that one semicircle begins with 
a greater velocity than another." (Nic. Oresmius, Episcop. Lexo- 
viensis) (Lisieux). ' 

From the SAME to the SAME. 



1 7, CAMDEN-STREET, CAMDEN TOWN, 
* May 3, 1852. 



* Your questionless answers received, and yourself thanked for 
same. The roots of an equation determine an equation ; but your 
answers do not determine the questions in some cases. I can, 
however, make it out well enough. 

' It reminds me to return you the proof [of questions proposed 
at the examination for Bishop Law's Premium]. Your questions 
are good ; but do you examine so exclusively in matters of geome- 
try and its algebra ? Your school is too much tending towards 
curves and surfaces and nothing else ; for which you and your 
pupils will in the next world be turned into long thin snakes, with 
an intuitive apprehension of the equation of the curve you happen 
to be in for the time being and this for ever. You will twist 
about to try to get your own equation out of your head, were it 
but for a moment, without any success. You will be separated 
into branches united by one consciousness, that is, by the fei-lin^ 
that there is but one equation to both ; for your personal identity 
will be the consciousness of having an equation. Think of this 
while there is yet time for repentance.' 



VOL. in. 2 A 



354 Correspondence between Sir W. Rowan Hamilton and 



From SIR W. R. HAMILTON to A. DE MORGAN. 

1 OBSERVATORY, May 3, 1852. 

* You are an authority on the subject ot functions; but I have 
no need to exhibit deference in the present case, for I feel the force 
of your reasoning. I had not the least hope of my own ever 
discovering even one value of (0 + 1)4, where and ;// are arbitrary 
functional signs ; nor indeed do I see any prospect of interpreting 
usefully log. * or sin * in every one of the many ways which may 
be imagined to exist. But quaternions, you know, are not en- 
tirely arbitrary operators. Besides their distributive and associative 
properties as multipliers, they have this in common with the old 
imaginaries, that each is accompanied with a conjugate, such that 
the sum and product of the two are what I call scalars, and are 
subject as such to all the old and usual rules. 

[He then determines the differential of the square root of a 
quaternion as a case of the solution of a linear equation (the 
work is to be found in Lectures on Quaternions, p. 628 . . . 631 
. . . 635), and concludes with a sketch of the subject of 603 . . . 
605 in the same volume.] 

* At last I make up your Walker, and remain/ &c. 

From the SAME to the SAME. 

' OBSERVATORY, May 5, 1852. 

* Thank you for remembering my wish that you should return 
me the little paper of questions. I was not sure of procuring any 
other copies, but have since been promised some (on paying for 
them, as I suppose). 

* I am glad to have the opportunity of saying a word or two 
about one of those questions, and that old plague, priority. No 
one can well imagine, on looking over the printed paper, that I 
claim anything for myself, except the quaternion analysis employ ol. 
Quite lately, as part of the new matter which I cannot resist th 
temptation of adding to my Lectures, in consequence of the d< 

of publication, occasioned by what, for the honour of this Univer- 
sity, I should be ashamed to plead to the public a want of pr< >; 



Professor A u gust us De Morga ;/ . 355 

type for the Contents I have written out, with some slight com- 
ment, the substance of those questions, to exemplify the "Ca! 
of Variations in Quaternions." Now, on looking over portions 
of your Differential ami Integral Calculus yesterday, partly to 
verify and profit by your recent reference, I lit on page 443, where 
not merely the remits are given, for a shortest line on a surface of 
revolution, but the same mechanical aerification (or explanation) 
assigned which had occurred to myself. I have accordingly 
pencilled the following memorandum, which I will or will not 
insert, exactly as you choose, when the proof slips come to hand : 
"A similar remark is made by Professor De Morgan, in his Diil'<>- 
rential and Integral Calculus." For my own part, I cannot say 
that I like to be, on all occasions, " mentioned." I was not pleased 
at Bronwin's citing, a few years ago, in the Catnbridijr ami Dublin 
Journal, as " Hamilton's Theorem," the very simple formula, 



whereas I had published, in 1831, this much more general one, 



If I had not the opportunity of consulting yourself, I should think 
it more respectful to you to omit, on my own responsibility, the 
above-mentioned reference to your " Calculus." But I believe 
that if I do print my little " Evolute Investigation," I ought ex- 
pressly to refer to you the property of the evolutes of curves on a 
sphere. You are aware, for I thiuk that I mentioned it, that my 
< lateral quaternion formulae for evolutes in space are not of very 
recent growth ; but the application of spherical curves was lately 
suggested by one of your many papers. 

* As to variety of topics, you may have noted that my printed 
list professed (with truth) to be merely a selection from my questions 
of the last three years. There were several other questions, about 
definite integrals, and various other parts of analysis, which I have, 
for the present, suppressed. I am only one of three examiners, 
for Bishop Law's Mathematical Premium, and do therefore the 
less injustice if I take occasionally a peculiar line. And you must 
remember that I am at present a Propagandist. If I could only 
make you a proselyte to the quaternions of course ! Meantime, I 
am, &c. 

2 A2 



356 Correspondence between Sir W. Rowan Hamilton and 

* P. S. As to variety in ray own reading, I hope to indulge- 
largely in it, when my book is off my hands. I mean, in science 
chiefly ; for, as it is, I do not quite neglect literature. This morning, 
for example, finding myself a little fatigued with mathematics, I 
read, in the open air, before breakfast, the whole text of the second 
volume of the Curse of Kehama, and some of the notes. I was not 
quite pleased to find, at breakfast, that both my boys failed to inter- 
pret, what their young sister did for them, on their translating for 
her the words, this somewhat cramp dictum of Aristotle, disbeliever 
in Birds of Paradise, which is quoted by Southey (from Henry 
More) in a note: on TTTTIVOV povov ovStv Icmi/, WCTTT^ vtvaiKov 
IJLOVOV <mv i\0v. 

* My copy of Kehama has \\Qos (in two places) ; but I do not 
remember meeting that form. Your books are among those which 
I hope to attack, as a reader, not as a controversialist. Did Henry 
Crabb Robinson ever tell you that I am slightly acquainted with 
my Edinburgh namesake ? I was introduced to him, at his own 
wish, in 1830, and took tea one evening at his house. He also 
pressed on me a sort of indefinite loan of a curious volume of 
modern Latin poetry, including a long and to us interesting poem 
by Leibnitz. But I have never exchanged with Sir W. H. even 
the slightest written correspondence. Had I done so, I should 
not have felt easy in corresponding so freely with you.' 

From A. DE MORGAN to SIR WILLIAM HAMILTON. 

' 7, CAMDEN -STREET, CAMDEN TOWN, 
May 6, 1852. 

' Your letter received also Walker, for which many thanks. 
I see that John Walker could think about logic as well as about 
arithmetic. 

* Neither Walker nor myself has any claim on account of 
merely incorporating the negation into the predicate. The change 
implied in passing from "man isnot horse" to "man is not- 
horse" is as old as Aristotle, or nearly. What I believe I claim is 
the introduction of two new forms involving contraries, namely 

" Some not X is not F," 
" No not X is not Y," 

\vhioh cannot be transformed into Aristotelian assertions or nega- 
I about X and F, mean : ng, in truth : 



Professor A ugu sfus De Morga ;/ . 357 

the first, " There is which is neither X nor Y " : the second, 
" There is nothing but is either X or Y." 

1 The introduction of these and their systeraatization with the 
rest I have never found anywhere, before or since I wrote. 

1 Your dealing with d . r* may be quite right. But it is to be 
remembered that partial interpretations may solve problems. We 
got on well with the positive square root before the negative one 
was known. Liouville (I think) solved problems and explained 
difficulties, with fractional differentials, based on 



So did Peacock in the Report, based on 
d m .x n Tn + 1 



But their two systems did not agree. 

' Thursday. Yours received. I would rather not be cited for 
so little a matter as the revolution surface shortest line. I agree 
with you, and rather dislike citation on small matters. You can 
cover yourself by " as has been noted," if you like. 

' The theorem 

A) . *, 



in the old form, is HerscheFs : see Examples of Calculus of Diffe 
rences. You stand where the separator of 



.fx, 



stands with respect to Taylor's theorem. The other, more general 
one, is yours, I have no doubt. 

* Citation leads to queer results sometimes. At the La 
College in Bedford-square (where my daughter goes) the t- 
of mathematics quoted me about logarithms in his class ; and my 
daughter heard one of the girls saying " Oh, I wish Mr. De Morgan 
had never invented those logarithms." 

' Why should you have felt uneasy in corresponding with me 
if you had done so with Sir W. Hamilton ? / correspond with him. 
When he sent mo his book (a few days ago) I wrote him a 
ending with the following alternative : " I hope you will by ... 
prevent my having recourse to the knife, and leave me to cut you 



358 Correspondence between Sir W. Rowan Hamilton and 

up with the pen as occasion shall serve." And instead of binding 
me over to keep the peace, he consented to my stipulation. The 
matter was this: when my " Formal Logic " was published I sent 
him a copy as his right, he being replied to in it ; and he, like a 
petulant schoolboy, sent me back the book, because he was offended 
with the dressing I gave him. So, when he sent me his book, in 
a small part of which (thirty pages) I am mauled, I gave him to 
understand that I had sent him my book as the right of a person 
attacked, and that I looked upon the thirty pages as my right, and 
that if he did not accept two works I sent him (arithmetical books 
and almanacs), I should feel obliged to cut out my rightful thirty 
pages, and return the rest. Fill up my blanks above . . . with 
"accepting my opuscuia," and you see that it will not read so 
murderously as might be thought. 

* I cannot find ix^oc in the dictionary at all, and doubt it being 
a Greek word. I see you trouble yourself to write accents. I find 
scholars beginning to get rather sick of them, and I hope to see 
the day when they will disappear, as also the soft breathing '. I 
did not even know you knew Crabb Robinson. He was at my 
house yesterday evening quite hearty. 

* As you are doctrinally with the Church of England you can 
ask a question which, from such a heretic as I am, would be flat 
blasphemy. 

The Greeks, we all know, fought about three i/Trooratrac in one 
ova ia. 

{/TToaratnc ouata. 

substantia essentia. 

The Latins have made it three personae in one substantia. What 
proof is there that t/Trooraatc in Greek ever meant persona, or ovaia 
substantia ? 

'The place in Hebrews where uTroorarr/e is translated j0ir*0fl is 
in \apaicnip THC viroaTaaewQ. Now xo/ocncrTj/o never meant itna<i> 
translated : it was a mark, particularly a graver's or sculptor's mark, 
as in a seal. "Why is not this " the impress in his substance" instead 
of the "express image of his person " ? When you meet with u 
theologian, you may put him to his defence. 



Professor Augustus // ui. 359 



From SIR W. E. HAMILTON to A. DEMORGA 

* OBSERVATOBT, May 8, 1852. 

1 1 shall, in the first place, get off my hands (with your per- 
mission) the question you raise, about a part of a verse near the 
beginning of the Epistle to the Hebrews. On turning to a copy 
of the New Testament (London, 1850 ; Taylor, AValt.-r, and 
Maberly), professing to contain Griesbach's Text, with the 
various readings of Mill and Scholz, I find, in the first place, that 
the text of the passage (Heb. 1. 3) is not marked as dou 
We have therefore, I suppose, the same Greek words in view. 
Now, I am well aware that \apaKTrip and ayaX^a are not int. r- 
changeable words. (I avail myself of your proposal to omit 
accents.) But still I cannot think that x a P aKTr lp T> 'C wTroorao-tojc 
signifies " the impress IN his substance"; even if the words were not 
here followed by the pronoun aurou, on the soft breathing of which, 
as printed, I should lay no stress, if it did not appear to me to be 
contrasted, by the whole context of the passage, with the avrov 
which very shortly follows it. Merely from the Greek, I should 
infer that the vtoe, the (cA^po^o/ioc, was the IMPRESSION 01 the 
UTTOTacnc of the 0coc XaXrjo'ac rote TrarpaGiv. The word \apaKTiip 
is used, I believe, occasionally, with a genitive, to denote what 
we might call the characteristic of some thing or person. But 
when referred, less figuratively, to the notion of a stamp, or brand, 
or graver's impression, I imagine that x a 9 aKTr i9 TIVOS signifies 
the stamp impressed by, or the one originally belonging to, rather 
than that received by, the person or thing of which the name is put 
in the genitive ; or that the TIQ is the giver, not the receiver, of 
the impress. The context in the verses referred to appears to me 
decisive on this point, HERE, even if my recollection should be 
wrong, as it easily may be, in what concerns the general usage of 
the word.' 

From the SAME to the SAME. 

' OBSERVATORY, May 8, 1 

* I wrote to you this morning a note, of this size, as to sheet, 
on xapaicrrjp, &c. "Whether I shall forward it remains to be seen. 
You would neither like it as an agreement with your views, nor 
dislike it as a rude opposition to them. But I may think, on 



360 Correspondence between Sir W. Rowan Hamilton and 

looking over the note, that it had too much the air of scholarship ; 
whereas I cannot pretend to much more than a liking for the Greek 
language, and an early and tolerably accurate acquaintance with a 
comparatively small number of Greek writers ; increased since as 
to extent, but doubtless diminished as to accuracy, by subsequent 
pleasurable but careless hours of reading. 

1 About the Walker's Logic, I think that after my first exulta- 
tion, on getting in a small Dublin shop a book which I supposed 
you to have sought for in vain, had evaporated, I wrote to you to 
mention, or confess, that I feared it would not interest you so 
much as I at first hoped; and that I suspected the point about 
"not-mastiffs" was not your point. But you must now be 
informed, if indeed you are not convinced of it already, that I 
have hitherto read your "Formal Logic" only as an entertaining look. 
A story goes, that a person, who read more than he digested, once 
told a friend of his that he had heard people talk of " Euclid," 
and that he was curious to read the work. The other lent him a 
copy, and was surprised to find the borrower return it the next 
day, with many thanks. " "What, have you read it ? " said the 
lender. " Yes, thank you," replied the borrower. " Eead it ? 
read Euclid* through in that short time ? " " Oh, if you mean the 
A's, and B's, and C's, I skipped all those" The " Quaternions," as 
well as the " Formal Logic," may meet with some readers of that 
stamp ! 

' But, seriously, I do not despair of yet reading your book, or 
rather that book of yours, as well as some others. When your 
"Algebra " reached me, I forthwith attacked the " Introduction," 
and read it regularly through. (I was amused to observe that my 
son, William Edwin, your acquaintance, despised, or at least 
.s7-////*v/, the said introduction, as if it could convey no new infor- 
mation, or instruction, or suggestion to him, and went on, at once, 
to the First Chapter ; so that, on comparing notes, he was found 



Long ago I was on a visit at the same house, with a most ladylike, but not 
vt-nj int 11. cttial Englishwoman, who, happening to hear some of the party talk 
one evening about the obscurity of some parts of Coleridge's * Aids to K< tKr- 
tion,' requested to be allowed to take the book to her room. At breakfast, the 
next morning, she said she had read it through, and could not imagine \vln-n; 
the difficulties lay. To be sure it was summer time, and aho may have been an 
early riser ! 



Professor Augustus De Morgan. 361 

to "be far ahead of me ! He is no prodigy, but a good and sensible 
and reasonably industrious boy, and may yet do very well.) I also 
spent a good while, soon after receiving them, in looking over, with 
much interest, and some amusement, your " Arithmetical Books," 
and " Book of Almanacs " ; and was engaged, for two or three 
hours to-day, in reading the early parts of your " Differential and 
Integral Calculus," which acquire a new interest to me just now, 
from my researches on differentials of quaternions. On that sub- 
ject (differentials) I must write to you again/ 

From the SAME to the SAME. 

1 OBSERVATORY, May 9, 1852. 

' I shall let the note about x. a P aKTr i9 g> but nave no wish nor 
ability to engage in any controversy about it. It was in August, 
1850, that I met Henry Crabb Eobinson, at Eydal Mount. Wo 
were afterwards, for a while, fellow-travellers by railway. He 
may remember reading to me, out of my copy of "Wordsworth's 
poems, the lines addressed to himself. But it is not unlikely that 
he may have quite forgotten that we ever met. We had a little talk 
about Sir William Hamilton, whom I had then lately seen. 

* Did you ever see the enclosed Abstract, of date December, 
18-10? It contained the extended form of Taylor's Theorem, 
which I had already applied at that time. Are you sure that the 
little theorem, 



is not mine ? Just glance at sections vii. and viii. of Herschers 
E.nnnples again. I am sure that he shall be most welcome to the 
restitution, if the property, such as it is, be his own. 
' Is Herschel still at 32, Harley-street ? ' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEN TOWN, 

' J/i/y 10, l,v 

' As to HerschePs Theorem, which you have trnn>lutrd into the 
calculus of operations, without knowing what he had done, look at 
my Differential Calculus," pp. 307, 308. Your/(l + A) . gives 
his expansion. In one sense they are different theorems. 



362 Correspondence between Sir W. Rowan Hamilton and 

'I don't remember seeing this particular Abstract of 1845. 
But it reminded me that years ago you forwarded to me a good 
bundle of Abstracts to distribute, with instructions, however, not 
to distribute many till I had some further instructions from you. 
These have never arrived. And I have only distributed a few. 

* As to uTTooracnc you gave me, with brief interval, two infor- 
mations, one written, one printed, of your " Church of Englandism,"" 
as Bentham calls it. So I tried a pass with you, as you gave me 
the measure of your sword. But I am no lover of controversy. 
Remember, I have no doubt whatever that aurov refers to 0toc. 
The question was, whether viroaraaiQ could be rendered person. I 
maintain it to be a something of his substance. Some of these 
days I will dilate on the mode I intend to employ, when I have 
time and money, to apply the theory of probabilities to the ques- 
tion whether the anonymous epistle to the Hebrews was written 
by St. Paul. 

* When you tell the story about Euclid, remember that the man 
left out the " A's, and B's, and C's, and the pictures of scratches 
and scrawls." I once met a man, no strong mathematician, who 
said he read Airy's Gravitation through on a bench in the front of 
his house. 

' When you read my " Formal Logic," take the Paper in the 
Cambridge Memoirs the second one first. It will give you all the 
pith and more.' 

From SIR W. B>. HAMILTON to A. DE MORGAN. 

4 OBSERVATORY, May 12, 1852. 

'/was the aggressor there is not a doubt about it you could 
not avoid paying me back a little. Let me at least plead, oh the 
score of politeness, that I did not intend firing off two barrels at 
once. When I picked up for you, in December last, the Walker's 
Logic, first edition, which I had never seen, nor indeed heard of 
as an edition, before, and saw, in print, prefixed, the author's st ; 
ment of his departure from the Church of England, I felt mv 
impelled, out of a sort of honesty, to make a memorandum on the 
page, that I did not agree with him. But the result was, that I c ; i 
to feel a sort of delicacy about sending to you the book at nil. 
should probably have tried to let you forget it altogether, but 



Professor August us De Morgan. 363 

pur reminding me lately of my promise. My answers to the 
Boyal Commissioners were drawn up, after long delays, in 
haste, "Haw Haste, half-sister to Delay" (Tennyson), an<l 1 
suppose that their only non-faulty element is their veracity. 
Whatever led me to insert the statement of my " Church of 
Englandism " in them, when I sent you that unfiual copy, I was 
merely thinking of the circumstance, that we were freely commu- 
nicating for the time, and that I had received proof sheets from 
you (of your triplets), as well as sent you others. I request you 
therefore to believe that, in point of fact, I did not wish to provoke 
you into an argument, although I grant that I may very natu- 
rally have seemed to do so. The subject is far too grave for me to 
treat it lightly ; nor do I think that you desire to treat it so. It 
involves no less than this : To whom are divine honours to be 
paid? I agree with you (as I suppose), in blaming the ordinary 
Bonian Catholics, especially the poorer ones, for worx/tfjtjting her, 
whom yet I agree with them in calling the Blessed Virgin. The 
question, as respects myself, and the Church in which I was bap- 
tized, and to which I belong, amounts to this, at least: Are we 
(the members of that Church of England and Ireland) idolat> 
worshipping Christ ? in praying, not only through but to him ? 
That is the only question which / think important in the matter, 
but to which, partly, no doubt, from influences of education 
although as little, I think, from influences of interest, as in any 
other human case I answer without any doubt; asserting my 
(confessedly partly taught) belief, that we may pray to Christ ; and 
that, of course, we therefore ought to pray to him. You mav, if 
you can, overthrow the received, or (so-called) orthodox, interpreta- 
tion of the word viro^ams, and it will concern me very little. Y n 
may establish the old doubt of the Pauline origin of the EpistK- to 
the Hebrews. I am not a particularly timid reader, and shall 
read your publications, on those as on other subjects, if I ha\.> 
access to them. But I am not very likely to adopt your general 
theological views, and have no hope of your adopting mine, 
it only remains that I should repeat my regret for having exposed 
myself to the charge of having sought to draw you into a c< 
versy which was not likely to be profitable. Who knows bu' 
we may yet have some useful controversy on some matheni 
subject?' 



364 Correspondence between Sir W. Rowan Hamilton and 



From PROFESSOR DE MORGAN to SIR W. B. HAMILTON. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
'May 16, 1852. 

* Imprimis slip 92 : De Morgan not de Morgan when I 
was at Cambridge, I used to get out of my misery in riva voce ex- 
aminations sooner by M D than I should otherwise have done, 
by insisting on this capital arrangement. 

' Herschel still lives in Harley-street, as before. He is some- 
what of an invalid, but I think is better. 

'I do not fix on you an attempt to make a controversy. It 
was I who took advantage of your profession of faith to see 
whether you had paid attention to all the meaning of your profes- 
sion. But you have subsided into general adherence for a mem- 
ber of the Church of England who does not care for the meaning 
of vTToorao-tc is a great deal nearer to me than is Dr. Pusey, for 
instance. When I came to think about my M.A. degree, I found 
I should be required to declare that all who dealt wrongly with 
substance and person would perish everlastingly ; and so I continue 
B.A. 

* If I were to enter upon your modified question of the worship 
of Jesus Christ, I should have to ask whether you mean worship in 
the English of the seventeenth century or of the nineteenth. If 
in the sense of the man who fell down and worshipped his master, 
or of the worship of the sciences which Baptista Porta's trans- 
lator speaks of, or of the worship which James I. declared to belong 
(only) to gentlemen, when he forbad them honour* (in an edict in 
which he rates justices of the peace for allowing themselves to be 
styled your honour when their proper title was only your irorx/n'p) 
to take the three first instances I remember THEN Mahomet did, 
and Socrates would, worship Jesus Christ ; and that worship, at 
least runs from one end of the New Testament to the other. If 
you take the modern sense, I should fall back on the practice of 
the apostles. I am content to go as far as they did. N. B. tl. 
alterations of meaning rudely confuse minor distinctions. The 
icorship of idols was forbidden, i.e. the minor mark of resjuvt 
a fortiori the major. 

* I first saw this cited in one of Theodore Hook's novels : he used to rake 
up such things. 



Professor Augustus DC Morgan. 365 

* The inclosed is a puzzle for your daughter. 

'We want a name for that collection of n a points, real or 
imaginary, through which pass an infinite number of curves of the 
n th degree. Is not a poriswfitw .sy/.sVrw the proper name ? Given 
n z points, to pass a curve of the n th degree through them. Gene- 
rally impossible in certain cases one system sometimes ttco, but 
tht-n an infinite number. This last is precisely the Greek porism 
to the best of our knowledge of it. 

1 1 am now going to turn my attention to writing an article on 
the case of my friend Libri a man whose persecution will be the 
great blot of French science for many a day. The means which 
have been taken to stifle all discussion of the case in France will 
give it, I hope, all the more currency in the rest of Europe.' 

From the SAME to the SAME. 

'7, CAMDEN-STREET, CAMDEN TOWN, 
'-May 21, 1852. 

' Keceived, among other things, your scrap* declaring that \/~\ 
is of multiple interpretation. I talk so little to mathematicians 
that I really do not know what they think about controverted 
points. With the exception of Libri, I have not, since our cor- 
respondence on quaternions began, spoken so many words to a ma- 
thematician on mathematics as I have written to you nor to all 
mathematicians they are few and far between and I have no timo 
to cultivate their society. I should have said, Is it possible any- 
body can hesitate about this multiplicity ? For example, there is 
an infinite number of interpretations deducible from mere double 
algebra. Let a certain plane (A) be that of ordinary double 
algebra ; and let another plane (B) be that, say, of transference. 
Let every point in space determine a curve ; that is, let the point 
(a, b, c) determine the curve 

x = (v, a, b, c\ 

y-\l> (v, a, b, c), 

* = X (*9 rt > ^> C )> 

say the curve of transference (projection). Let a point in A have 
a corresponding point in B, determined by the curve of t: 
ference through A. Then algebra interpreted in A by double 

* [This scrap is missing-. 11. T. 0. j 



366 Correspondence between Sir W. Rowan Hamilton and 

algebra may be interpreted in B by the transferred or projected 
lines. Though generally speaking */~\ would depend for its 
meaning upon the point to which it is referred. 

The passage from anything to its meaning would involve a 
000' 1 kind of operation. I remember, when I was writing on 
functions, I wanted a name for this species of interrupted inversion 
<t>0<j>- 1 , and I asked Whewell for one. He answered instantly, 
" a sandwich of course," which I thought ingenious but inapplicable ; 
though the likening of inverse operations to bits of bread and 
butter turned contrary ways was a pretty analogy. 

* I very much want a short and sound demonstration of 



rt 
.1 



x m dx 



sin [m + 1- 

V n 



The common one is sound enough, but long and too unartistic. 
* If for x we write 3? (p positive) we get 



rn-o 

of which any percentage may be got into by making p large 

Jl-O 

enough ; so that (p =00 ) it is an instantaneous integral, and a may 
be infinitely small. But I do not see any simplification in this 
and here I leave it. 

' I saw this morning, for the first time, the original publication 
of Format's method of maxima and minima, tangents, &c., in a 
volume of Herigone's Course of Mathematics (1644). It is another 
instance of the manner in which a person's first ideas will smell of 
his predecessors more than his later ones. His increment (c) is 
an absolute 0. Let e = 0, substitute a + e for a in 0#, &c. This 
was Kepler's view. "The infinitely small increment," says he, 
" absolutely vanishes at the maximum or minimum." 

From SIR W. E. HAMILTON to A. DE MORGAN. 

'KoYAL IBISH ACADEMY, May 24, l.vvj. 

* I have just been told that I am likely to be called on to : 
some account this evening of a subject begun by me some time n 



Professor Augustus De Morgan. 367 

Continued Fractions in connexion with Quaternions and /ttquater- 
nions. The title of this subject was not printed as part of the bill 
of fare for this evening's meeting, and I came in to Dublin not at 
all expecting to come on. It will be well if I can remember the 
leading formulae, or at least the leading points, in time, especially 
as I am going, in half an hour, to dine with the Academy Club, 
which I do not often do. So I shall merely add at present. 1 
that when I said I was sure you were right about multipli 
interpretation, I was thinking rather of 



than of </^i ; for I agree with you that the interpretations of it 
are probably infinite ; 2nd, that I have read your recent proof of 

dx 



and think it quite satisfactory.' 

From the SAME to the SAME. 

* OBSERVATORY, May 26, 1851. 

* Do you know that I am not sure that I may not meet you 
next week at Greenwich ? For some years past the compliment 
has been paid of inviting me to be present at the Visitation, and to 
dine (paying, of course) with the Visitors afterwards. The annual 
card arrived by post this morning, and I am tickling my fancy 
with the notion that this time I may act upon it. If so, I suppose, 
we shall be for an hour or two together (on June the 5th). But 
if so, I must return, perhaps that very evening, without any 
opportunity of paying my respects to any of my acquaintances in 
London. It must be, as concerns myself, what the " Herschel 
Dinner" at the Freemasons' Tavern was in 1838, when I ju>t 
went to London for that one purpose of attending it, and returned, 
the purpose being accomplished. 'Tis true that I was ; 
enough to visit St. Paul's, and to climb into the Ball where the 
strange effect of the wind made me think that it was " the sighing 
of the Heart of London." Something to that eii'oct I wrote that 
evening, on board a steampacket, to the late Marquis of Northamp- 
ton, with whom I was on what might fairly be called intimate 



368 Correspondence between Sir W. Rowan Hamilton and 

terms : with the present Marquis I am merely on terms of a dis- 
tant civility. (He was not at Castle Ashby when Lady Hamilton 
and I made our visit to his father there, in the autumn of 1838). 
But about the " sighing "am I to quarrel with Dickens, or 
Dickens with me, because he printed almost the same image or 
figure in one of his publications of a later date ? Where is this 
priority business to end ? I am as sick of it as you can be ; but 
still, in anything important as regards science, I should take it as 
a favour to be warned, if I were inadvertently exposing myself to 
the charge of plagiarising. As to verse-writing, I remember 
copying for Lady Rosse, in 1848, a sonnet, such as it was, which 
I had composed by starlight in the highest gallery of the great 
telescope at Parsonstown ; and noting that one line resembled some 
verse of a living poet, to which I was, just then, unable to refer- 
The only tolerable part of my sonnet was the conclusion, namely, 

" Pursuing still its old Homeric march, 

Northward, beneath the pole slow wheeled the Bear ; 
Rose overhead the vast Galactic arch ; 

Eastward the Pleiads, with their tangled hair ; 
Gleamed, to the West, far seen, the lake below, 
And through the trees was heard the river's flow." 

'Now, as to the " tangled hair" of the Pleiads, I cannot swear 
that this may not have been, in some dim and half -conscious way, 
suggested by the " Locksley Hall " of Alfred Tennyson, at the 
time forgotten by me. He has, if I now remember for I won't 
stir to look for the book something like this : 

" Glittered like a swarm of fireflies tangled in a silver braid." 

' Shall I dash boldly, before I stop, into some new verses of my 
own ? They are certainly not worth offering to Mrs. De Morgan, 
fur whom, without prejudice to her title to receive, whenever I can 
recover them, some more of my old baby lines, I am about to 
a couple of sonnets of my own; and to whom I request yoi 
present them. These are for your own amusement, and are now to 
be written from a remembrance of what I extemporised, b>t\\ 
the acts of shaving yesterday, for the entertainment of a brother 
rhymester, . . . 

'The verses, then, that I thus dashed off, on the spur of the 



Professor Augustus De Morgan. 369 

first jingle between the sounds of splashing and /, were an 

account, for the said brother versifier, of a dinner of the Royal 
Irish Academy Club, on Monday last (the Queen's birthday) i 
a flag was flying, in honour of that day (May 24), from the top 
of Nelson's Pillar ; and they were these if I can remember 
rightly : 

' I helped the soup without any splashing, 
Also the salmon without much mashing ; 
Cut up the fowls in morsels nice, 
And served about the melting ice ; 
And, seeking every taste to please, 
Scooped out the ripe and rotten cheese. 
Then, rising from the Chairman's seat 
But first our Dean had blessed the meat, 
And in our names due thanks had given 
For all the boons of gracious heaven 
I called on all the social Board 
Their hearts and voices to afford, 
With loyal and with glad acclaim 
To welcome a majestic name ; 
And, on our sovereign's natal day, 
In cordial, earnest, toast to say : 
Long be her years, bright and serene ! 
Yictoria's health ! God bless the Queen ' ! 

' That the decanters did not circulate too fast, nor too often, 
may perhaps be inferred, in my case, from the circumstance that I 
was called on, that evening, with scarcely any notice, and spoke 
for about an hour respecting the ^/quaternions, receiving all sorts 
of compliments afterwards from the President (Dr. Robinson) 
and the Academy. Seriously, they open a new world of difficulty, 
for some future Alexander to conquer. It is honour enough for 
me to have indicated the direction. 

' I must tell you that my daughter behaved very well yesterday 
though, just before the operation, she whispered to me that she 
should like to be put asleep. She had been given her choice. You 
would have mesmerized her ; I was content with chloroform, and 
the dose given was not enough to prevent her from huv 
sort of dreamlike consciousness. She is doing very well. My 
cousin Hut ton is a skilful surgeon. 

* I walked out from Dublin, after the evening meeting more 
than five miles. The night was fine.' 

VOL. III. 2 B 



370 Correspondence between Sir W. Rowan Hamilton and 



From A. DE MORGAN to SIR W. R. HAMILTON. 

* 7, CAMBER-STREET, CAMDEN TOWN, 

'May 27, 1852. 

' I shall not be at the Greenwich dinner, which I shall much 
regret, if you are to be there. My lectures are so' late that I can- 
not undertake to be there. And in truth there I have never been, 
nor ever was I at a meeting of the Royal Society, nor at the 
British Association. In fact, I am the most ungregarious animal 
living. 

* My wife will be much obliged to you for the verses, which 
are very pretty, and I say ditto for the dinner rhymes. But you 
cannot keep to Dean Swift's vein, you walk out of it into loyalty, &c., 
and it was a mercy you did not amble into sonnet metre, and give 
one the idea of an inversion of " desinat in piscem mulier formosa 
superne " ; setting out with Swift's head and shoulders, and end- 
ing with the feet of one of the Muses. 

* I am very glad your daughter is out of it [a slight operation 
near the eye], I hope for good and all. Where chloroform can be 
ventured upon, it is more speedy than mesmerism, which requires 
some previous trials. There is a kind of consciousness left by both, 
very often. The first man operated upon (amputation of the leg) 
by mesmerism in England said "he felt a crunching" it was the 
saw / " but it did not hurt." ' 

From the SAME to the SAME. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
<J/ay31, 1852. 

* You get your answer by return of post for this is Whit 
Monday, and here I sit holiday-making, literally puzzling myself 
or rather having puzzled myself with the following question, 
for two minutes, by mere forgetfulness of geometrical extension : 

* How is the end of the shadow of this pole* to be determined ? 

* On looking into several books of perspective, I see that they 
take care not to mention the case always having the sun high 
enough to avoid it. 

k llow does your daughter get on ? ' 

[The shadow of a high pole cast by the setting sun.] 



Professor Augustus De Mo) . 3 7 r 



From SIR W. E. HAMILTON to A. DE MORGAN. 

'OB8ERVATOBY, 1852. 

1 . . . Herschel wrote me a very kind letter lately, saying 
lie was so glad to see the " Contents/' as a token of approaching 
publication, and reminding me of the shortness of life. I do re- 
member that ; but so possible is it to apply a motive in two ways, 
that the remembrance makes me all the more anxious to bring out 
now what I hope (you will smile) to publish this very month, as full 
an account as other claims on my time will allow of the state to 
which the quaternion calculus has been already brought. 

* This morning I rose about six ; took some tea standing, for I 
was too busy to sit down ; deciphered some half-effaced characters, 
which I had chalked upon a blackboard, as lecturing myself, a 
few months ago, on the subject of a new sort of variations o> 
uitc integrals (in quaternions, you understand), not depending on 
any infiniteness of ike function between the given limits of integration, 
and therefore not anticipated, so far as I yet see, by Cauchy. 
your Calculus, or his Memoir, perhaps a rare one, on definite 
integrals between imaginary limits (Paris, August, 1825), which, 
if you have not got it, I shall with pleasure lend you, besides 
returning your Mourey but wish to have both a little longer. In 
that Memoir, at page 57, Cauchy refers to a formula of "M. 
Gruillaume Libri, dans le tome 28 des Memoires de PAcademie des 
Sciences de Turin." Is he your illustrious and unfortunate friend ? 
Something, but very little, too little, I know of an eminent Libri's 
researches, for instance, respecting linear differential equations ; 
but have none of them at hand to refer to. 

1 To descend to myself, I went after the lately mentioned 
deciphering, and after getting something copied for me, on my 
car to Dublin, with my eldest boy, procured cap, gown, and band 
at the place where such things are kept for me near the College ; 
arrived at the large and handsome hall, or theatre, devoted to 
examinations, elections, &c., and in which the new Provost 
other members of the Senior Board were assembled, for publicly 
conducting the usual Fellowship Examination ; proceeded to the 
seat in the inner circus, which Brink! ey used to occupy, and to 
which 1 had successfully vindicated my claim, by the single pro- 
cess of staying away, when, about twenty years or a few more ago, 

2B2 



372 Correspondence between Sir W. Rowan Hamilton and 

Provost [Bartholomew] Lloyd (an excellent man) excluded me- 
from it, for a year or two, on the grounds that I was too young, 
and too much at home in this my native University, not to sit 
with the Junior Fellows ; heard the statutes and oaths respecting 
the election read aloud in Latin by Dr. Todd, and was amused by 
his involuntarily falling back occasionally into that language after- 
wards, when, novelty of novelties ! he proceeded to examine in 
English on what is here called " Logics," including Aristotle, 
Kant, Hobbes, Priestley pray admire the chronological order ! 
Locke, Stewart, Brown, Reid, Lord Monboddo, and Sir William 
Hamilton. (There were others ; but I missed De Morgan.) How 
strange it sounded to hear all this discussed in the vernacular ! 
1 Would M. Libri accept a copy of my book when published ? * 

From A. DE MORGAN to SIR W. R. HAMILTON. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
'June Sand 9, 1852. 

' In your remarks upon the result of substituting x + y </^\ for 
x in $x = 0, and the n* - n supernumeraries, you give a meaning to 
<l>(x> 2/) = 0> a 8 the equation of a curve, in double algebra. This I 
conceive to be bringing the uncircumcised into the inner court of 
the temple, for which Paul got nearly stoned, I think ; but we live 
in more quiet times. 

1 When x and y are real, they have no interpretation, except 
upon the unit line. 

'A quantity of common algebra is only quantity. One of 
double algebra is both length and position. Every determinate 
symbol, when it is measured from a determinate point, symbolizes 
a determinate point. 

' Let us now mean by small letters lines on| the unit line ; 
and let large letters be quite general. 

' What then is this circle ? * Answer, the locus of 

x + v/^7 2 . v/^l [' > **]' 

According to conventions, every point here signified is on the cii 
every point not here signified is not. 

[Here U inserted diagram of a circle with radius a, central i 
axes, and an ordiuate corresponding to an abscissa #.] 






Professor Augustus DC Morgan. 373 

1 If you like, on the unit line, to take y = ^/a*-x*, and then to 
give v/ 2 - a? its quarter-turn, you may. 

But y-o/a^x* [*'<*] 

is only interpretable in the unit line. 
1 When & > a\ then 



x + -y/ - x 2 . *- 1, or x + v* a - a a 

is all in the unit line. 
' Accordingly, 

x + v/v - x* . <y - 1 

is 1. [V<0 2 ]. The circle any point in the circle 

2. [# 2 >fl 2 ]. Any point in the unit line got in one particu- 
lar way. 

''Example. It is required to divide 2x (in the unit line) into 
two algebraical parts, whose algebraical product shall be a 2 (in the 
unit line). 

' Ansicer. x + ^/x~ 2 , and x <v/# 2 a'. 

' Here it is plain ' [from consideration of the figures] 'that 
a problem has its locus in the circle and the unit line conjointly : 

circle for x < a, 
unit line for x > a. 

But now we ask what does 



represent ? Answer, if X be perfectly unlimited, any point 
whatsoever, but only as obtained in one particular way, or set 
of ways. 

' Under what restrictions is its locus the circle ? The question 
here is, given a curve and its representation, required the co-ordi- 
nate system under which the given curve shall have the given 
representation. 

*Any expression may belong to any curve, if the proper 
co-ordination be adopted. 

* In common algebra, length only is represented, and position 
understood ; consequently, only permanent positions can be kept 
in thought as co-ordinate. 



374 Correspondence between Sir W. Rowan Hamilton and 

' But any two lines which start from the origin, and end at a 
given point, may be co-ordinates of that point. 

'If X=x^~\ T=y^- [ be the co-ordinates of a point, the- 
system of co-ordination is defined by two equations : 



These equations are, in the common system, 

-o, ,-* 

which we bear in mind, but not as expressed conditions. So i 
plane geometry, my pupils bear in mind, but not expressly, the 
second equation of all the curves, 2 = 0, that is, they always keep 
in the plane ; but when we come to solid geometry, they always- 
find it difficult to remember that their curve in xy is now expressly 



= 0. 
Let r, be the common polar co-ordinates of a point. Then 

z |vr 1 + ^v:] = ; , <v--i 
or 

x cos + y cos r\ = r cos 9 = x^ 

x sin + y sin r\ = r sin 9 = y\. 

If y\"\ x i be the common equation of a curve, we have then. 
three equations to the curve, as we ought to have : 

*(*,y, ,n) = o, (i) 

*(*,y,5,n)=o, (2) 

a?8in + y8in?j = x(#cos + 2/ cos?)). (3) 

The common three equations of a straight line are 



Ilence, given one of the four, length or direction of one co-ordinat\ 
the others are found. 

'If X be given and ^ and ^ unknown, it is a functional 






Professor Augustus De Mor^ in. 375 



equation with two unknown functions to make a given equation 

w(*,y, 5,n)= 0, 

combined with (1) and (2), give (3), which is the problem of mak- 
ing a given equation represent a given curve. 

* But what do we get when we generalize x and , &c.? Only 
a covert way of choosing another system. 

'If x = x'+ x"*/^\) &c., our co-ordinates are 



and $ = i// = become four equations. But we ought to have 
six. If we take six equations at pleasure, between x x", &c., we 
have a seventh in 

x sin + y sin ?j = ~%( x cos ? + y cos >j), 

where <r, &c., are functions of the eight. But this can be reduced 
to the first considered case for the six equations of co-ordination, 
together with the four which express #, &c., enable us to eliminate 
x x", &c., and leave us two equations between x y r/. 

' What then, you may ask, do you mean, in double algebra, by 
the equation (x a y} ^ in its most general sense ? I reply that, 
first, you have no right to make real values of y perpendicular 
to those of x, unless you expressly mix two systems of double 
algebra, as D. F. Gregory and Walton have done in Cambridge 
Mathematical Journal, and as I shall do to a greater extent in the 
next number. These duplicate double algebras do good service in 
interpretation, but do not mix in one algebra. 

( To interpret, for instance 

**+y*-l, 

or rather X*+Y*=1. 

First, Jand Y being real, each solution symbolises a pair of points 
in the unit line. 

'Next, x being x+x"</~\, and y being y' + y"<S^\ 9 each siu-h 
solution symbolises all the points derivable from 



To make this give a curve, we must have recourse to definite co- 
ordination, as above described. 



376 Correspondence between Sir W. Rowan Hamilton and 

* If we want something more like the theory of curves, we 
must make some hypothesis such as Gregory has made, e.g.. Let y 
be measured in a plane perpendicular to that of #, its real part 
being the \/^T part of x. Then, for the point Q, 



The equation (X, F) = 0, then belongs to a surface, and 
y Y) = 0, to another. All the intersection of the two surfaces 

does not belong to the system j ? ~ Q, but only those points of it 

which are symbolised by the same co-ordinates. Thus the point Q 
may be on both surfaces, but may not be the property of both 
equations for the same co-ordinates being on one for 

OM + MN^~\ and NP + PQ . -/ry, 
on the other for 

^ and N'P + PQ . -/^l. 



' I foresee that some of these days partial solutions of equations 
will be considered. For example, given <j>(X 9 Y) = 0, \I*(X, Y) = 0, 
required the solutions of the two equations which agree in the real 
parts of both, or in the imaginary parts, or in the real parts of 
one and the imaginary parts of the other. 

' This gives complete interpretation and objective existence to 
the nm points of the intersection of two curves of the m th and n th 
order which is all that is wanted for full investigation of the 
various cases. 

* Your aspiration after some new world to conquer in double 
algebra I cannot join in. I have no objection to any extensions ; 
but when every operation is actually explained, I have what I want. 
You want a geometrical instrument well, you have made one ; but 
algebra arithmetic in the widest senseis not made for geometry. 
There is a law of thought to consider ; if, upon a preconception of 
what we wish it to do, we make it an instrument for a special 
purpose, we divert attention from our main object, the consideration 

hat law of thought. I hold that double algebra must remain 
as the full development of the conceptions of arithmetic, so far as 
those symbols are concerned which arithmetic immediately suggests. 
The invention of other symbols is legitimate. 






Professor A it gust us De Morgan . 377 

I The great feature of the complete algebra is geometry intro- 
duced for explanation of algebra not algebra adapted to geometry 
and the great use of the extended field of explanation is t i 
algebra the science of quantity by symbols we m/ ^//, t hr- >u _ 

I never hope to see the day when there will not be a science of 
symbols in agitation, in which extended power < 
the thing we are in quest of, but the leaves of the ledger which 
^are made up are not, I hold, to be reopened, though new accounts 
may be opened. 

* Are you sure that if you were to go on with your bicouplet* 
you would not land in quaternions, which I have always hoped 
would turn out to be the legitimate triple algebra, or algebra of 
space. 

I 1 have a clearness, as the quakers say, that double algebra is 
in and of a surface, not necessarily plane; that single algebra 

true linear double the true areal and that a real triple algebra 
is to come. I shall rather quarrel with nature if your quaternions 
with the three rotations be not the triple algebra. This is the 
point I want to investigate. 

*I did once try a kind of bicouplets. Seeing that a and b 
when understood go into a + b*/~i, where */~i is a new kind of 
symbol, I thought the next step would be to imagine 

a + &V/TI + (a' + b'^/~\) . A-, 

where k is some new kind of quantity. But I made nothing of it. 
All the laws of algebra were exhausted, and there really was 
nothing for k to do that I could give it. 

1 Now you have tit for tat (what does this phrase mean ?) as to 
quantity. And I intend to set to upon some old notions about 
discontinuity and the principle of mean values. 

* To-day it is making up for the drought. It has rained 
incessantly for eighteen hours and more. 

* Are you much of a politician at elections ? I never gave a 
vote in my life.' 

From SIR W. E. HAMILTON to PROFESSOR DE MORGAN. 

1 OBSERVATORY, June 12, 1852. 

' Thanks for your printed paper, and long note, both of which 
reached me this morning. Ten days (or nights) ago, I wrote to 



378 Correspondence between Sir W. Rowan Hamilton and 

you a double-sheeted note, but laid it aside with the purpose of 
adding to it, and now not that I pity you on that account there 
seems but a poor prospect of its ever turning up again. For you 
must know that I gave a sort of official dinner in my Library on 
Wednesday last, as being then (for the 26th time) one of the 
Examiners for Bishop Law's Mathematical Premium, and was- 
obliged to allow housemaids to use pretty freely their discretion 
in clearing out the room. Orders were given, no doubt, to abstain 
from destroying papers ; but to all practical purposes, many, 
indeed most, of those which were lying about are hopelessly 
hidden from my view ; unless, indeed, I shall be roused, after my 
book is out, to "take stock," as tradesmen call it, in various ways, 
as regards volumes, manuscripts, observations, &c. The last are 
not my property ; but I fancy that of my papers a vast number 
must be not worth preserving. Still whatever are preserved may 
with advantage be classed. 

* June 14. I write in the printing office, and have not your 
last note beside me, which I regret, for there was something in it 
that I wished to answer, as indeed there have on several occasions 
been, when I could not write till it seemed too late. Let me at 
least mention that I have just signed 2 R, in which are just intro- 
duced, with brief explanations and proofs, some results respecting 
that sort of variation of a definite integral, arising from non- 
commutativeness of multiplication, in virtue of which (as I think 
I mentioned some months ago) we have, in quaternions, such 
formulae as 

> 

or more fully, 

- 

where 

^ = 0, ^ = 0.* 

That subject I have finished, for the present, and have written out 
some investigations respecting differentials of functions of quater- 
nions, more general than that respecting the differential of a square 
root. 

[An example is then given, to be found in Lectures on Quater- 
nion*, 634, p. 630]. 

See Lecture* on Quaternions, 627 (p. 621). 



Professor Augustus De Mor 379 



1 1 shall be happy to be allowed to present you with two or 
three copies (or perhaps more) of the book, when completed, for 
Undistinguished scientific friends of yours, or pupils not yet known 
to fame. As to scientific men of celebrity, I need not say t 
shall take it as a favour, and indeed a service, if you will assist me 
in making out (soon) a list of such as presentees. // I can 

go in that way I am not yet certain ; but am aiming to be liberal/ 

From A. DE MORGAN to SIR W. R. HAMILTON. 

' 7, CAMDEN- STREET, CAMDEN Town, 
'June 15, 18-VJ. 

* The matter you refer me to in 2 R (slip) is new, I have no 
doubt. I don't see who is to get on the ground, except yourself. 

1 The parties I remember in Great Britain who are likely to 
read your slightly abstruse book are, in England Herschel, 
Peacock, Babbage, Whewell, Leslie Ellis (Camb.), Stokes (Camb.), 
Graves, Sylvester, Cayley, Spottiswoode (Oxf.), Kirkman, Young 
(Havil, Great Southampton-street, Camberwell), O'Brien (King's 
College, London), Wheatstone (who is curious about such things). 
I wish you would send one to Libri. I will answer for the 
charges against him being malignant in their character ; and, so 
far as definite, fully answered by him. I hope to send you a few 
notes on his case shortly, in print. He is at Florence House, 
3, Chepstow Villas, Bays water, London. 

1 In Scotland, Kelland (Edinb.), Wm. Thomson (Glasgow). 

' 1 dare say there are more who will come into my head. But 
I do not live among the mathematicians, and new ones start up. 
So no more this evening.' 

From the SAME to the SAME. 

' 7, CAMDEN-STREET, CAM DUN TOWN, 
* June 20, 1SJ'-'. 

' All the people who can read the quaternions are sure to be 
of name ; and therefore it is difficult to propose a person who 
would not come into your own a priori list. Do you know Dr. 
Logan of Prior Park? He is a person who could and would read 
it. I forget whether I named him before. He has the history 
of mathematics for 1700-1852 better than any one I know. 

* Libri is the writer of the Paper you mention, and a great many 



380 Correspondence between Sir W. Rowan Hamilton and 

others. Besides being a first-rate mathematician, lie is par excellence 
the historian of mathematics who knows the manuscript sera. His 
history of mathematics, &c., in Italy, of which four volumes are 
published, is a work which will live. 

' His is a lot with a good many alleviations. All the bibliogra- 
phical world throughout Europe is not only convinced of his inno- 
cence, but of the extreme malice and ignorance of the accusation. 

' Imagine John Smith accused of stealing a certain book, printed 
at London, as appears by the title-page ; in which book, on the 
fly-leaf, is pasted the book-plate of John Jones of Manchester. 
Imagine our Attorney- General English or Irish insisting on it 
that John Smith pasted in the book-plate of Jones of Manchester to 
persuade the world that the book was printed at Manchester, and 
not in London. This is imagination ; the following is reality : 

' In Libri's possession was found a book printed at Venice, as 
appears by the end to which everyone looks in an old book for 
place and date. At the beginning, in the usual place, was stamped 
in the name of the convent of St. John of the Canals at Piacenza, 
to which the book once belonged. The experts (or professional 
bibliographers) employed to examine Libri's books declare that he 
stamped in this book plate to make it appear that the book was 
printed at Piacenza. They then point out that Venice is at the end, 
and wind up with " De tels faits ne se discutent pas il s'ex- 
posent ! " 

4 It is a considerable alleviation of such a charge that the 
knaves who conduct it are such fools. Another is that a lady to 
whom he was engaged and a very charming woman did not 
forsake him, but came over to England and married him. Another 
is, that this lady has about 1500 a-year ; and her daughter, by a 
former marriage, has or will have as much more. Another is, that 
hb own property is in Tuscany, where the French cannot get at it. 

' You probably know Mr. Kirkman, who extended, or rather 
augmented, Pascal's Theorem. He has just published a book of 
mathematical mnemonics, the most curious crochet I ever saw. 

* Some of these days I will try to write a catechism of mathe- 
matics in which every answer shall be remembered by a double 
meaning, e. y. \ 

1 Q. What is a rational state of things ? 

' A. One in which all radicals are exterminated. 



Professor Augustus DC M 



'Of course you will not forget the following libraries : The 
scientific societies, the colleges, and Sion College, London, 
you may never have heard of a clerical library the librarian is 
a good mathematician ; Advocates' Library, K<linlmr<:li ; Observa- 
tory Library, Greenwich : a fair one and in high order. A 
the prince of methodists. You and I should look very small before 
him. My theory is that when he tries his pen on blotting-paper 
he makes a duplicate by the pressing machine, files, an<l indexes 
it. When he wanted communications of advice and suggestion 
about the altitude and azimuth instrument, I sent him an 
from Jupiter, warning him not to follow his (Jupiter's) dan. 
about. It was read to the visitors, and filed with the rest/ 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSEBVATORY, Wednesday Morning, 
June 23, 1852. 

* I intend to make many presents, including foreign ones ; you 
are aware that I wish to give a few copies, through you, to promis- 
ing pupils or other private friends of yours, who are not known to 
the public as writers. It is not my expectation that for the other- 
wise published copies there will be an extensive, much less a rapid 
sale ; but with all its faults, and with all my belief that, if it were 
now to be done over again, I could write a much better book, I still 
think that it will always be regarded as a sort of classical work, in 
one department of mathematical science, which shoots forth ahvu<lv 
ramifications to almost every other department, and from which 1 
have also strong expectations, not likely to be realized 1)}- m 
as regards electricity, magnetism, and all that depends on polarity . 

' Your story about Airy is capital. 

' Mr. Kirkman has sent me his book it is a curiosity perhaps 
it may be useful to new men; but I suppose I am past using it. li 
he can make me remember the formula, 

cot a sin c = cos c cos B + sin B cot A, 

which I am forced to copy now, for I have vainly tried, for (to 
speak moderately) thirty years, to commit it to memory, he will 
have achieved a feat. Yet I can repeat long strings of lines 
Milton, Wordsworth, and others, not to mention all, or i. 



382 Correspondence between Sir W. Rowan Hamilton and 

the sonnets that I ever composed and you are far from knowing 
their number. 

* You asked me lately whether I took any interest in politics. 
I am not an ardent politician, and most unaffectedly distrust my 
own political sagacity, and warn my hoys that I do so. It is likely 
that you and I would be found to differ, without bitterness, on 
those, as on theological subjects. But since you give me the 
opportunity, I shall add that, on the principle attributed to Solon, 
I have always voted at every contested election, in the College or 
city of Dublin, since I had a right to do so ; and hope to assist 
soon in getting rid of Reynolds. (Who was it that said, " I have 
no prejudice, but I hate a Frenchman ? ") I have just received a 
circular, requesting me to vote for the present members (Hamilton 
and Taylor) for the county of Dublin, which I shall certainly do, 
if I find that I have the right. Hitherto I have always imagined 
that my tenure of this Observatory did not give me a vote in the 
county. And now perhaps I have said enough about politics. I 
may just add that I would not have made any exertion to displace 
the last ministry ; but in voting I feel as if I were on a jury.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

'July I, 1852. 

1 1 want to lodge the following somewhere by way of publication^ 
a word you understand I use in the primitive sense. There is no 
mistake which introduces so much confusion into scientific history 
as confounding publication with printing. It is only a little phase 
of algebra applied to logic. . . . 

[Here follow fourteen pages occupied with a treatment of syllo- 
gism considered as a composition of relations, &c. It concludes 
characteristically thus :] 

1 And so much for genus and species. Now for a play upon 
words. No doubt genus always possesses species ex vi terminorum. 
To prove that it is not lawful to take a letter out of a word, and 
insert it in another, proceed as follows. If possible, let the second 
in the second word be changed into i and inserted in the first. 
Then, genius always possesses specie. But this is absurd, there- 
fore, &o. Q. E. D.' 



Professor Augustus De Morgan . 3 83 



From the SAME to the SAME. 

' 7, GARDEN-STREET, CAMDEN TOWH, 

'July 11, 1852. 

* The heat here has been so oppressive that men, women, and 
children are dried up into mummies. If you have anything here- 
tical in mathematics, or orthodox in theology, to insist upon, now 
is your time ; I can oppose nothing. 

' Chasles has published one volume of his Geometric S" 
have not had courage to look into it yet. Do you know any- 
thing of (1) Liouville's edition of Monge ; (2) Poinsot's Theory 
of Kotation (1851) ? 

* There is, of course, no hope of news from a mummy as I 
claim to be, and to remain till we have a thunderstorm, or some- 
thing to alter the air. I was amusing myself the other day with 
-a work on logic, or rather against logic, written by one Justin 
Brenan. I suppose he must be an Irishman. Do you know 
Anything of him ? ' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

'OBSERVATORY, July 12, 1852. 

1 Though I don't much like copying, even from myself, I think 
it may entertain you to see a copy of part of a note whicli I have 
just written to that cousin of mine, who learned, when she was a 
child, and when I was able to teach such things, some Hebrew, 
Persian, and Sanskrit, on my lap. Perhaps also it may be less 
impolite to you to send the gossip in this form, than if I were to 
address it directly to yourself, who probably take a different side 
in politics. 

* " My dear Bessy, after thinking and writing about the di'qua- 
ternions on Saturday morning, and giving William Edwin [your 
London acquaintance . . . parenthesis addressed to A. De M.] a 
long lecture on practical astronomy, I shaved an exertion ! and 
went to Dublin with him on my car, from which we got oil . 
walked as we approached to Green-street. A strong force of 
police preserved quite enough of order. I am not sure that I 
should not have enjoyed a little more of a rote [pronounce like 
noic], and I easily found the proper tallyroom, and a person to 



384 Correspondence between Sir W. Rowan Hamilton and 



show me the way to the proper polling booth. The following ques- 
tions may not be reported verbatim ; the answers are nearly so : 

* " Q. Under what title, or by what number in the Registry of 
Freemen do you claim to vote at this election ? A. 1399. 
Q. (After inspection of the Registry.) Are you the person re- 
gistered under that number ? A. Yes. Q. Have you already 
voted at this election ? A . No. Q. For whom do you vote ? 
A. Grogan and Yance. (A voice] Thank you, sir ! 

' " And so ended my little part in the important drama of the 
day. I walked about Dublin a great deal with Will [your said 
acquaintance], and part of the time with Sydney [a sister of 
mine], and thought that my native city looked really very lovely. 
The Liffey was pretty full, which makes all the difference in the 
world between its beauty or deformity ; and the sun was shining 
on it, and on many beautiful buildings. Even the reflection from 
house- windows, towards evening, had a fine effect. (Some win- 
dows were smashed, by- the- way, as I was slowly driving, about j 
half -past five, on an open car through a crowded street ; but that 
was to be expected.) I availed myself of my first free day, since 
the Moore Meeting, to leave a card at Charlemont House." 

1 The remainder of my note to my cousin relates to money and 
other private matters. I have heaps of things to write to you\ 
about, and in fact have at least two rejected or suspended notes to 
you, thrown aside, though perhaps not a bit more foolish than this 
one, and I may yet take courage to send them. Will you bel: 
that I have only just discovered that the Bi- tensor of a product of 
Bi-quaternions is = the product of the Bi-tensors ? ' 

From A. DE MORGAN to SIR W. R. HAMILTON. 

< 7, CAMLEN-STREET, CAMDEN TOWN, 
July 14, 1852. 

* Whether I am for or against Grogan and Vance I can't toll : 
1 . Because I have not the least idea of their politics ; 2. Because! 
I have not the highest degree of certainty of my own. I huvo no 
objection to give Lord Derby a chance. I should bo glad to think 
that he or anybody else could get through luxincxx. I should likr 
to reduce the House of Commons to its original function of grant- 
ing supplies and stating grievances, leaving the executive 
remedy them on principles to be settled by the House. I fort 



Professor Augustus J} c Morgan. 385 

that the present machine will break down under its own incom- 
petency. I am glad you have ki-k<-d out the ol e man. 

1 1 admire the innocence of your mode of information that 

row rhymes to now. Let the circumstance be fairly stated. 
Irishman writing from Dublin of an election, admits that ho v 
have been glad of a row ; and thinks it necessary to inform his 
correspondent how the fourth emphatic is pronounced. Not ex- 
actly so, perhaps, for I understand that the parenthesis about 
pronunciation was intended for me. You did not, I presume, 
think it necessary to add that little bit of instruction in Irish to 
the Hebrew, &c., which you formerly gave your fair cousin. 

* There has been no contest in my borough, and I did not vote 
when there was one. I hate the system. Given two persons of 
whom I know nothing; required which is the best qualified to 
manage matters of which I know next to nothing. The presump- 
tion is that 5000 incompetent persons, by a contest of opposite 
incompetencies, will produce a competent decision. This absurdity 
fills the House of Commons. Another, as great, the House of 
Lords, namely, that the son of a legislator must be fit to be a 
legislator. And this works well, on the whole. I have long been 
of opinion that, in English hands (not objecting to a mod 
mixture of Irish and Scotch) , anything would work well ; and 
that Parliament as now constituted proves it. I would not hav.- 
all Irish, because the progress of events which should be in a row 
meaning one after another, with order and method, rhyming to 
slow might be subjected to a mispronunciation. Nor would I 
have all Scotch, for, when they had their own matters to manage, 
there is no denying that they mispronounced quite as much as the 
Irish. It was only, in all time, when they had to work their way 
among foreigners, that the Scotch were the decent quiet people 
which we know them to be. Are you of Scotch origin ? by name 
and arms you seem to be so. 

* Macaulay used to maintain that the House of Commons must 
have been the beast of number 666. The 658 members, by his show- 
ing, + the number of officials requisite to the action of the House 
I forget who they were made exactly 666 just taking in the 
clerks, sergeant-at-arms, &c., essential to its records, the enforce- 
ment of its orders, &c. 

* And so much for my politics. 

VOL. III. 2 C 



386 Correspondence between Sir W. Rowan Hamilton and 

' I shall send you, in a day or two, a separate copy of an article 
in Bent ley 1 s Miscellany on the Libri case. There never was such 
a mess as these unfortunate French have got into. 

* There is another light of Italian mathematical history just 
beginning to appear in England. I received yesterday, from 
Prince Boncompagni, the person in question, three tracts on 
points of Italian history. One of them is on Gerard of Cremona, 
the first translator of Ptolemy from Arabic. It contains (among 
other things) an Arabic algebra, translated by Gr. of C., in which 
there is notice of negative quantity, a thing hitherto wanting 
in all Arabic algebra. This is a step towards the connexion of 
Arabic and Hindu algebra. The latter abounds in negatives/ 

From the SAME to the SAME. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
< July 16, 17, 1852. 

4 1 have found the value of oo . You are required to admit 
three things : 1. That 89 60' = 90 ; 2. That tan 90 = oo ; 
3. That any man, having a point left incomplete, may make it up j 
by ancient authority. 

* If you boggle at the third postulate, I refer you to your j 
namesake at Edinburgh, and his followers, who never hesitate at j 
imposing a law of thought as necessary, if Aristotle has declared : 
it. I am not joking. If you or I can see no necessity they refer ' 
you to Aristotle to prove it. 

* Supposing you now satisfied on the third head, I refer you to | 
Reinhold's Tabula Directionum, 1554, in which is the first canon 

fcecundus, or table of tangents, that ever was carried to single 
minutes. 

* The last degree is to every 10 seconds. Now I quote 

89 59' 50" 206185567010 

89 60' 10000000000000; 

oo = 10 13 . 

Everything has its reason. Here is a kind of companion to 
tutu* *//JM. Since 10000000 was the greatest sine, and it \\ 
found constantly requisite to divide 10000 ... by a sine; ami si) 
the same 1000 . . . has often to be divided by a tangent, l: 
seems to have thought that a total tangent would bo a <l 






Professor A itgiisfit / >, . 1 f, -;>, / . 3 3 j 

able thing ; and so he put down 10000 to as many ciphers as the 
row would take. 

* I had written most of the preceding when your note came in, 
wherein June 23 appears as date. If you had sent it at onoe I 
would at once have given you my mnemonic for the formula you 
mention. The clue is 

cos -f- sin = cot. 

Now choose your beginning, say 

a c. 
Make a cycle of pairs 

a c c b b a. 

Change the last three into angles 

a c c B B A. 
Now, cos + sin = cot. 

Put cosines in the middle, sines on the flanks, and cotans on the 
extreme flanks, 

a c c b b a 

cot a sin c = cos c cos B + sin B cot A 
cot sin t cos cos t sin cot. 

This is the only formula for which I have ever been obliged to 
invent a mnemonic. Mr. Kirkman says his is better than mine. 
I have not looked at it you may. All I know is that I never 
forgot the above rule from the time when I first made it. And 
in mnemonics the difficulty is always cu&todire cmtodon. 

' Could you not make a sonnet out of the formula ? There 
arc just 14 elements, 

cot a sin b = cos b cos C + sin C cot A 
12345 6 7 8 9 10 11 12 13 14. 

* This evening we have rain and thunder. And as Sir Sydney 
Smith (the Acre man) insisted on it to a friend of mine the largo 
rain-drops knock down the wind, and make it blow in the lower 
regions of the atmosphere. 

' Pray do not write, and then consider whether you si i all send 
it or not. If I were to do such a thing you would not get all you 
!get out of me. 

1 July 17. I have been hard at work all day valuing the 
iffairs of an assurance company, and incidentally committing high 

2C2 



388 Correspondence between Sir W. Rowan Hamilton and 

treason imagining the death of the heir to the Crown ; inas- 
much as the Prince of Wales has had his life assured by someone, 
and, the rate of interest being 3 per cent, if you take the premiums 
at anything less than thirty-three and one- third years* purchase, 
you imagine his death. 

1 But I opened this letter not to expose you to the hazard of 
misprision of treason, but to tell you that I send a copy of the 
Libri article with it. You will now see how the land lays. 
There are a great many more facts, but I had not room for them. 
And all that I have stated I have as good reason to know can be 
proved as it is possible I could have.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, July 20, 1852. 

' 1 received this morning your letter of July 16, 17, and also 
your pamphlet about Libri, which I have read with great satisfac- 
tion, but intend to read again. And I do not think that I thanked 
you for the one of July 14, which promised me that pamphlet, and 
contained some words about politics. On this last head I do not 
think that we should differ much certainly not so as to prevent, 
or render difficult, a good-humoured correspondence, now and then, 
upon the subject (whatever might be the case with Theology). 

* About a year ago I happened to meet, in Mrs. Charles 
Graves's drawingroom, a Dublin lady of some talent and a b/neish i 
tinge, who set herself to draw out my opinions on various points. 
Among other questions, or half questions, she said, " You ar 
Liberal in politics, I believe." " Eeally," said I in return, " 1 
do not know exactly what the word means ; but as it sounds 
intended for a compliment, I suppose that I had better accept it." 
Yet I regularly voted against O'Connell, and that not only wlim 
Lord Normanby (at whose hands I received knighthood) \\ 
Viceroy, but also when his co-candidate, against whom I voU-d 
also, was Eobert Hutton, now of Putney Park, near London, 
whom probably, or at least perhaps, you know, and who is my 
maternal cousin in the second degree. I must say that Eol 
1 1 utton, on the occasion I have in my mind, came up to me as I 
was waiting for my turn to vote, and, knowing well what I intended 
to do, shook hands with me in public. About a week afterward* 
I received an invitation, in Dublin considered a " command/ 1 



Professor Augustus DeMorgnn. 389 

dine with Lord Normanby, and sat next Drummond you know 
perfectly whom I mean. We had met before, at the measurement 
of the base, beside Lough Foyle, in 1828, and had enjoyed a 
Sunday's stroll along the banks of that lake, in addition to dining 
together under a tent, and some other little incidents, extremely 
savoury to me at the time, of a semi-military life. In our after- 
el inn or chat at the Viceregal table, at that later period of which 
I spoke before, we did not shrink from politics, although in some 
important points we differed about them, and the subject of the 
ballot was mentioned. Whether Drummond was for the ballot, I 
forget, but remember that I was led to say, ayuinxt it, k 
what satisfaction or comfort could I, as a gentleman, sit now at 
Lord Normanby's table, if I had voted secretly, instead of openly, 
against his candidates in Dublin a week ago ? " 

* My " liberalism," if an existent quantity, won't prevent me 
from voting, if alive and well, for Hamilton and Taylor, at the 
Dublin County Election, on Thursday morning next. I belong to 
the new sect of Derbyites ; but I trust that you believe that I 
would have voted exactly alike if Lord John Russell were still in 
power, namely, for just those candidates for whom I have voted, 
or intend to vote, at present.' 

From the SAME to the SAME. 

' OBSERVATORY, July 20, 1852. 

* Letters and pamphlet received more very soon from me. 
Guess, if you can, what you have to say to the enclosed receipt for 
Sir J. Hamilton ? I am the person meant, and the basket, with eight 
compartments a practical biquaternion is for holding letters 
from, and some few memoranda of letters to, you ! Already it has 
been, with great resulting pleasure, applied to the said purpose ; 
but a tradesman waits in my hall, and, while paying him his bill, 
I shall bargain for his posting this memorandum.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN Towir, 
'/H/y23, IMS, 

'You meant me to wonder at the receipt, and I did won 
till your note explained it. I congratulate you on your metho- 
dical habits. 



3QO Correspondence between Sir W. Rowan Hamilton and 



*I saw to-day, with great satisfaction, an article in the 
on Libri's case, of which article I knew nothing whatever. The 
article I sent you will be circulated by Libri himself, and copies 
will go to Ireland to a great many official people. I begin to 
hope that people in this country will be brought to see that there 
is no criminal justice in France. But there is no one to collect 
such jewels of criminal law as the following : 

' 1. Five men robbed a diligence, many years ago ; three men 
were convicted and guillotined ; some years afterwards three more : 
now 3 + 3 = 5 + 1, therefore one innocent man was guillotined. 
Which it was was discovered and fully proved. In fact, the 
second trial exonerated one of the first three. The family of that 
unhappy man cannot get his sentence reversed to this day. The 
Courts would not do it. The late House of Assembly would not 
recommend it. It was against the dignity of justice, they said, 
to reverse under such circumstances. All persons admitted that 
if he had been at the galleys the sentence would have been 
reversed, and he would have been set free ; but as the mischief was 
wholly irreparable, the dignity of justice required that the stain on 
his friends should continue. 

* 2. A man is tried for secret society work. Evidence, a police- 
man, who swears that certain respectable persons in the man's 
neighbourhood told him that the man was a Socialist. Prisoner's 
counsel require him to name these parties. The Court overrules 
the question. The man is convicted. 

' Nine men out of ten whom I meet with have more or less of 
idea that a refusal to go and stand a trial in France is an inevi- 
table presumption of guilt. One thing, however, I have learnt 
from all the remarks I have heard on this case, and that is, that 
the law of England is far above the average Englishman in its 
notion of justice, and that a study of the law would improve the 
loyi'c of nine out of ten. 

*I do not see your name among the office-bearers at Belfast. 
Do you go ? Belfast reminds me that the treatment of Young 
was more French than English or Irish either, except ui 
religious bigotry. I hope the Queen's Colleges will be free of 
this plague. 

' By the way, is it held true in Ireland that the priests educated 
at Maynooth ore more National and less Papist (in the lit< 



Professor Augustus De Morgan. 39 1 

sense of the words) than those not so educated ? I have heard this 
a I firmed, and made a reason for the continuance of the grant, and 
not by Roman Catholics.' 

From SIR W. R. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, July 23, 1852. 

1 You won't think it necessary, I suppose, to enter a formal 
protest against what I said (if I remember it), when I last (or 
lately) wrote on the subject of politics ; nor to disclaim the degree 
of agreement between you and me on that head which I assumed 
potentially to exist. You need not take pains to inform me that 
you are not a Lerlyite, in the sense in which I avow myself to be 
one: it is quite enough for me, that you state yourself to be 
" willing to give the Derby Ministry a fair trial," or that you 
have used words to that effect. We agree in preferring (ini- 
mt'iixurably, as I hope and think) England to France, and to 
Italy, as regards constitution and government witness the "Libri 
case." You could not, any more than myself, contemplate coolly, 
as an endurable thing, the conquest or the possession of the British 
Empire by Louis Napoleon, or by the Pope. You might probably 
prefer, theoretically, the absence of all alliance between Chun-li 
and State; may even wish and hope to live to see any such alliance 
dissolved, but would not choose to throw public affairs into cou- 
fusion, and paralyse, or at least obstruct, all vital actions of the 
national body corporate, by any such course of conduct as may, to 
judge by the past, be adopted in the next Parliament by the 
Brigade. You may be, in some very important respects, a 
Radical ; but, if so, your intellect, good sense, and good feeling, 
must oblige you to be (if I may coin the term) a liberal radl- 
\vhile I on my part claim to be (what I have been called) a 
liberal conservative. 

* My liberalism did, however (as I warned you), not prevent 
me, or rather, as by me interpreted, it urged me on, to attend at 
Kilmamham yesterday, and to vote there for Hamilton and Taylor, 
who, I am told, are certain to come in; but after an arduous 
contest for the Romish priests are understood to be actually 

nvassing a mild word for their mode of getting voters to 
poll ! At all events, Uberavi a it imam maim ; and of course I should 



39 2 Correspondence beiween Sir W. Rowan Hamilton and 

have voted for another Hamilton (GK A.), and Napier, in the 
College, if there had been any contest. 

4 1 write this note standing in the open air, and have just now 
(since writing the ahove) been told that Hamilton and Taylor are 
IN, by at least 500 ! Tar-barrels, &c. in this neighbourhood ! ! No- 
more common sense this evening ! ! ! 

' James Hans Hamilton I am long acquainted with, also with 
George Alexander Hamilton ; and though not so long, yet well, 
with Napier. But Grrogan, Vance, and Taylor, I am not aware 
that I have ever seen; and certainly I am not such a red-hot 
Protestant as Vance professes himself to be. But I act on the 
practical principle of taking the best I can get. If I had not voted 
for Vance, I should have helped to bring in Reynolds.' 

From the SAME to the SAME. 

1 OBSERVATORY, July 26, 1852. 

* Understand, first, that I don't pretend to be an unprejudiced 
man. Deeply prej udiced I know myself to be ; not thereby 
admitting that I am wrong. From childhood I have had political 
leanings, and always to the 71iberal side. My father (Archibald 
Hamilton, of Dominick-street, Dublin) was a liberal, almost a 
rebel ; he assisted Hamilton Rowan to escape from prison, and 
deeply involved himself by other efforts in his favour. On the 
other hand, his brother, my uncle, the Rev. James Hamilton, who 
lived for forty years the Curate of Trim, and died as such, was a 
Tory to the back-bone, and doubtless taught me Toryism along 
with Church of Englandism, Hebrew, and Sanskrit that last 
acquisition being pretty well lost by this time, although I still 
like the look of the letters ! My father used to enjoy the provoking 
me into some political or other argument, in which I always took 
my uncle's side. The seal, which I still use, and which contains 
a sort of abridgment of my father's arms, was given me by a 
maiden aunt, Miss Hutton, on the occasion of my triumphantly 
winning, as my father more triumphantly confessed to some frioiuls 
(or flatterers) whom he had collected, in a contest of eloqu* 
with himself for my first watch, which, when I was about tw 
years old, he affected to deny that he had promised me. Peril; 
before I finish this note, I may be able to lay my hand upon my 
father's seal, and to send you an impression of it. I should think 



Professor Augustus De Morgan. 393 

it pompous in me to me one so large and grand. We do clu 
be Scotch, of the time of James I. ; but believe that our pedigree 
can be traced back beyond the Norman conquest, to certain Earls 
of France. What nonsense ! I try to think so, practically, and 
am quite sure that I am glad that I am the son of a younger 
brother, and that if a dormant baronetcy in our branch of the 
Hamiltons shall ever be revived, it will be for a cousin, the present 
Rev. James Hamilton of Trim, son of my uncle above mentioned, 
and not for me. A poor knight is bad enough, but a poor baronet 
is something lower than ridiculous. 

* What set me off into all this rigmarole ! Oh, I remember 
now, you asked my opinion, as an Irishman, about the Maynooth 
priests, and I wanted to disclaim any pretence to being an un- 
prejudiced man. 

* An amateur genealogist asked me, about a year ago, what 
branch of the Hamiltons I belonged to. I answered that I had 
heard my father say, that he had claimed, and been allowed to 
claim, a distant cousinship with the Marquis of Abercorn ; but 
that I had never heard my father profess to be in any way con- 
nected with the Duke of Hamilton. " That is all as it should be," 
said the genealogist, " for the Duke of Hamilton is not a Hamilton 
at all.' 7 ' 

From the SAME to the SAME. 

* OBSERVATOEY, Monday Evening, July 26, 1852. 

1 1 should have no chance as a candidate for the City of Dublin 
(whatever might be the case with the College, if I were only such 
a fool as to allow myself to be proposed), because I could not 
promise to vote against Maynooth. On the whole, I am disposed 
to let the endowment remain. Such is the result of a very un- 
skilful attempt of mine to strike a balance between opposite evils. 
I am old enough to remember foreign-bred ecclesiastics of tlu 
Roman faith. They were extremely agreeable people, and much 
welcomed in Protestant society. The case is very different now. 
The education at Maynooth is, I believe, < //-English. But I 
repeat that I do not set up for being an unprejudiced man.' 



394 Correspondence between Sir W. Rowan Hamilton and 



From A. DE MORGAN to SIR W. B. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
'July 27, 1852. 

' I see you begin gemino ab ovo i. e. from Louis Napoleon and 
the Pope. Concedo. I do not want England to be governed by 
either. So far we go together. I am not sure I would destroy 
them. You have read " Pickwick." You remember Mr. Weller, 
sen., reasoning on final causes, says of death " What would the 
undertakers do without it?" Now I say "What would the 
newspapers do without " L. N. and the P. ? 

* Practically I hold as follows : People seem to be well satisfied 
with the Constitution, as they call it. / don't believe in the Con- 
stitution. I think this is the country of common sense, and we 
should make anything do. But, be the Constitution what it may, 
it is not my business to tinker it; for if it were I should not know 
how, except in a few small matters as follows : 



' 1. The Parliament not to be dissolved all at once, but =- = 94 

members to go out by rotation each year, and so that every man 
should serve seven years. The Queen to dissolve out and out, 
over and above, pro re natd. The rotational electors to be distri- 
buted through the year. 

' 2. No person to be allowed to go to the poll who shall not 
have given notice of his intention twelve calendar months before 
the election. In the event of no such notice given, the House of 
Lords to nominate 2x candidates for x to be chosen, out of whom 
the choice to be made. 

1 3. Every member of the House of Commons to swear, on 
taking his seat, that he has neither aided nor abetted bribery ; and 
to give in, on oath, a list of his election expenses all to be paid* 
before he takes his seat. 

* 4. Votes to be taken at the electors' houses, by papers left lo 
be filled up and signed (blank filling being permitted). These 
things would not suit the magnates who fool the mob septennially, 
but out of that raixuit dc plus. 



The House to shorten, at discretion, the term of service of anyone who 
pent undue sums, even honestly. 



Professor An gust it* J in. 395 

As to Church and State ; if the deadly poison of subscriptions 
be abolished, I should care little how they arranged their matters 
inter M. These subscriptions foster every kind of dishonesty. 

* As to the Maynooth Grant, if the Catholics settled down again 
into good subjects, I should continue it. And this because the idea of 
it being unlawful to countenance idolatry, &c., contains a prim-ipl" 
subversive of all things established. If the state conscience a 
creation of politicians must not encourage what it tl links false, 
how can the real private conscience be forced to do so? A Quaker 
has a much better right to refuse Church rates than the State has 
to refuse an allowance of public money, otherwise desirable, on tho 
ground of its disapproval of the doctrines taught in consequence. 
No more this batch. 

* I picked up the other day the tables of Eeinhold, containing 
(1554) the first table of Tangents (tabula fwcunda), published to 
minutes. He remarks that this table will be found very useful 
when one sine is to be divided by another, the two angles together 
making a right angle.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

4 OBSERVATORY, July 28, 1852. 

'I was much amused by your mathematico-political letter, 
which came to hand this morning. How many people do you 
hope to convince, or win over, to your new Utopia ? Though it is 
only fair to admit that you, like myself, seem to be rather glad 
than sorry, that neither of us is obliged to try his hcand, in practice, 
at " tinkering the Constitution." If a peer, or a peer's norniiu , 
before the Eeform Bill, had thrown out your suggestion No. 2, 
about the nomination of the 2o? candidates, what a popular clamour 
would have been raised! The "curry" business (Norfolk, 
it ?) would have been nothing to it. As to No. 3, against bribery, 
I am really delighted to think (what I have heard) that a recent 
Act of Parliament has almost anticipated your view, and mad 
crime nearly impossible. Yet I do not relish the thought of 
multiplying oaths. For my own part, I have on several occasions 
been called upon to take an oath, and trust that I have always 
done so solemnly and veraciously. But when, for instance, those 
questions about the Dublin city election had recently been put to 



396 Correspondence between Sir W. Rowan Hamilton and 

me, and I was again walking with my son William Edwin in the 
streets, I really had to reflect, for a short time, whether I had 
answered on my oath or not. The duty of veracity, at least when 
so publicly and solemnly appealed to, had appeared to me already 
so stringent, that no ceremony could increase it. The State, in its 
assumed capacity of conservator of public morals, has, I believe, 
assigned a special punishment to perjury. Most rightly, fitly (I 
admit), is that crime punished by any well-constituted society ; yet 
as a crime, not as a sin, by them ; although an awful sin it be. So 
I would amend your motion, No. 3, by substituting some such 
enacting words as these, " that a false statement on the subject of 
election expenses should be treated and punished as if it were 
perjury." But I repeat that I am told that a reform, with which 
I can thoroughly sympathise, has actually been of late effected, on 
this point of bribery ; and that the next House of Commons will, 
so far, be one of unprecedented purity. 

' At least I can say that it would surprise as well as pain me 
if I were to learn that a single person had been bribed, out of all 
those who lately returned, at three elections, the six Conservative 
members for Dublin/ 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
'July 31, 1852. 

*I shall not get any converts, for two reasons: 1. All the 
great parties would be against it ; 2. Nine-tenths of the electors. 
Who wants the people to have time to think ? 

' The proposal to allow the Crown to name candidates at a 
given time, if, by the laches of the constituent body, none were 
named before, would have made an outcry, which would have been 
answered immediately by references to cases in which a public 
trust placed in private hands lapses if it be not executed : e.g. if 
t lie patron of a living do not present within six months, he loses 
his right, and the bishop gains the presentation. 

'As to curry powder, which the poor Duke of Norfolk did 
recommend, as you state, I cannot make sonnets, but I have b< i 
now made epigrams ; and when the burst of public ridicule came 
on the Duke and all dukes some of the newspapers defended 



Professor Augustus DC Morgan. 397 

tli <-m by giving a list of dukes who had been in this and the 
battle. Whereupon the following came into my head : 

* Oh ! what than a duke can be prouder, 
Or who should be chanted out louder ? 
The dukes all in battle have stood, 
And because they have been food for powder 
They offer us powder for food. 

1 Don't betray me. 

' My plan would greatly diminish the quantity of oath-ta! 
Now, the candidates can have the bribery oath administered to all 
electors, and often do. At the same time I don't care for their 
having oaths. I only want the penalties of perjury, as in tho 
declarations which have been substituted for oaths in affidavits. 

* I got your seal, and the title, good for an advertisement. 
Your seal shows that between you and Adam come Scotchmen. 
How many generations of Irish ancestors have you ? 

' When you see the librarian of T. C. D., or any Irish biblio- 
grapher, professional or amateur, will you inquire wlu-tln-r tin TO 
are any copies known to exist in Ireland of the editions, first and 
second, of Lucas Pacioli's Summa di Arithmetica, &c., the earliest 
printed work on algebra. An Italian, Prince Boncompagni, who 
is writing on this work, is collecting all the accounts he can of 
separate copies. Nor is it unnecessary, for in the same edition of 
the same book three different first pages have been discovered, and 
there may be more. 

'The last day of letters prepaid in money. Henceforward 
Quot epistola, tot capita reyince, but not the converse.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

OBSERVATOBY, August 3, l v 

' For the satisfaction of an amateur astronomer of my acquaint- 
ance, I have spent some hours this morning on calcula 1 
extremely easy in principle, respecting the intersections of the 
orbits of some of the asteroids. If /,, and L m , denote the lati- 
tude and longitude of the northern intersection of the great-unit 
orbits of an m th and an n th small planet, and if Ceres, Pallas, and 
Juno, be taken as the planets 1, 2, 3, I find, using the elements 



398 Correspondence between Sir W. Rowan Hamilton and 

in Herschers large "Astronomy" (London, 1849, Appendix, 
page 647) the following numbers : 

/ 1J2 =10 9'55"'4; A,, = 187 47' 0"-7 (Ceres, Pallas). 
/,- 8 17 15-65; Z,, 3 = 209 49 31-0 (Ceres, Juno). 
7 2 , 3 = 38 7-1; 2 , 3 = 173 39 12- 1 (Pallas, Juno). 



' I won't answer Infractions of seconds, but shall be surprised 
if, so far as the mere trigonometry is concerned, any one of these 
results shall be found to be a whole second wrong, as a deduction 
from the above-mentioned elements. Slight changes in those 
elements might sensibly affect the results ; and I do not hazard 
any opinion as to the possible effects of perturbation, nor have I 
computed the radii vectores. But if my arithmetic be right, so 
far, it seems to be unfavourable to the notion of the asteroids being 
fragments of an old exploded planet. On the great circle which 
represents the heliocentric orbit of Juno, the northern intersections 
with the orbits of Ceres and Pallas seem to be distant from each 
other by an interval of more than thirty-six degrees. If I should 
find myself inclined, as a variety, to carry these little calculations 
any farther, do you think that the numerical results would be 
worth sending, to occupy a page, or less, of the Philosophical 
Magazine? [I suspect not.'] 

'You may, in the meantime, judge that I have no dislike to 
arithmetical or trigonometrical calculation, and that if I do not 
work more in that way than I do, it is merely because I imagine 
my time to be otherwise more usefully employed an opinion 
which the elder Struve urged on me as his own, and as that of 
other continental astronomers, at Oxford in 1847, in terms more 
strong than I should willingly repeat ; for though you may not 
believe it, I have a little modesty ! 

' Angmt 4, 1852. This is my 47th birthday, and my youi 
son Archibald's 17th. Without any deliberate pre-arrangvn, 
we were brought into the world by the same accoucheur, old Dr. 
Labatt (who was not the person engaged to attend my wife, but 
was sent out by him), within an hour or two of being esacf/y thirl v 
yean asunder ! There was some talk of calling him " Halloy," as 
tin; comet was just about to appear: at least his second name is 
Henry, of which you know the Falstaffian abridgment is Hal. 



Professor Augustus DC Morgan. 399 

He is a tremendous book-worm ; I consult his memory on all sorts 
of tilings. He is also a good boy.' 

[The second sheet of this letter is unfortunately not extant 
lost or destroyed. It probably contained Hamilton's account of 
his ancestry : see conclusion of De Morgan's letter of August 10. 
-B. P. a.] 

From the SAME to the SAME. 

OBSEKVATORY, August 8, 1852. 

* I can spare for Mrs. De Morgan, if you think it worth while 
offering to her in my name, the enclosed lithograph of the original 
letter, in which Wolfe sent to a friend of his the well-known lines 
on the death of Sir John Moore. 

1 Sheet 2 x will be finished in a day or two, and then there 
will remain only a short Preface, about which I must really c 
you. 

' I wish you would at once give me, as freely as you please, any 
general hints about what you think would be prudent and proper 
on the occasion even as to whether the first or the third person 
would be best to adopt only remembering that I do not 2" 
to do more than carefully to weigh any advice so given. Only 
part of a week remains to me, but I write quickly, when in the 
mood. 

' P. S. Lady H. and my daughter were saying, at breakfast 
this morning, that they should like to see De Morgan here.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

'7, CAMDEN-STKKKT, CAMDKX TOWN, 
' August 9, 1852. 

* Yours received both the old and the new. 

' The results you mention would be quite worth recording. I 
suspect the actual augmentation of the number of small pi; 
and \^Q possible augmentation, have rather a tendency to d 
the old idea of a planet split to pieces. Looking at the nebular 
hypothesis which will be very much looked at in the century 
coming the idea strikes one of a ring, nearly thick enough for 
maintenance, broken by some accident. If so, these little animals 



4OO Correspondence between Sir W. Rowan Hamilton and 

will swarm in a certain zodiac ; or perhaps, if Saturn's ring be not 
solid, but composed of very many and close satellites, we have in 
the asteroids the rudiments of such a ring. 

* I am at bibliography again, with the intention of giving 
something in the Companion to the Almanack next year. By way 
of a fair trial, I took down from my shelves the first four old books 
which presented themselves, binding myself in thought to make 
the trial, how much error and confusion would arise out of the 
four, either from the description others have given of them, or the 
description they give of themselves. I thought it probable that 

four books would yield two cases : they have yielded four, as you 
shall see in time.' 

From the SAME to the SAME. 

4 7, CAMDEN-STREET, CAMDEN TOWN, 
August 10, 1852. 

* My wife is very much obliged by the lithograph, which is very 
interesting. Now as to your points of difficulty. 

* As to the first or third person, I, for one, disgoze the third 
person (old Cambridge verb active, for "repel with disgust.") 
Tome 

' Segnius irritant animum demissa per " ille," 
* Q,uam qua) sunt " ego " subjecta fideli. 

4 1 disgoze also " we " from an author writing under his own 
signature. And our English is so entangled by want of genders 
and inflexions, that when "he," the author, and "he," the reader, 
get mixed up in a sentence, as they certainly will do, the clauses 
will change meanings, as Hamlet and Laertes did rapiers in the 
scuffle. No ! No ! keep I to take care of No. 1, and let " he " be 
the reader = " he or she," and let " it " be the book. 

'Your preface should, I think, give a concise and well-dated 
historical account, with a reference in intelligible bibliography, to 
everything you have published. This, with a total absence of all 
apology or deprecation in fact, an assumption of the rig) it to 
invent quaternions, to write on them, and to print the book IUMV 
be joined to anything you please that is not actionable. 

* I got your account of your ancestry : the following ma\ 
whether you have any Scottish feeling left. 



Professor Augustus De Morgan. 401 






[He then gives the famous brutalitt of Dr. Parr in reply to Sir 
James Macintosh about O'Quigley, ending with the words : " He 
was an Irishman ; he might have been a Scotchman."] 

' What will an Irish Hamilton say to the last clause P ' 

From SIR W. R. HAMILTON to A. DE MORGAN 

'OBSERVATORY, August 11, 1852. 

* This is my little daughter's twelfth birthday, and I hunted 
for your puzzle [square dissected into three squares] to amuse her ; 
but the solutions, such as they are, have been made out by my- 
self. Do you suppose that any third is possible ? The red ink is 
a birthday present from her, which I found on my pillow, in a 
parcel with pens, &c., when I awoke pretty early this morning ; 
that is always her notion of her birthday, that she is to make the 
presents. Her mamma found a pretty parasol. 

* Are these your solutions of the puzzle ? The pieces had been 
missing for some time, and only turned up to-day.' 

From the SAME to the SAME. 

' OBSERVATORY, August 13, 1852. 

1 You asked me lately whether I knew anything of Chasles* 
new work, Traite de Geometrie Suptrieure. He has had the good- 
ness to send me a copy, and I read this morning, honestly and 
fairly, the first six chapters through, to the end of page 80, not 
skipping anything, but, on the contrary, detecting several mis- 
takes of the press, not after all very important, nor perhaps 
worth noticing at all. 

* Besides regularly reading so far, I have skimmed much more, 
and even read with care some parts, up and down, near the end. 
I think I have now a perfectly clear notion of homogrnphlc figures, 
and of some other connected things. But what, as you may guess, 
more interested me just at present, was Chasles' view about 
imaginaries, and his application of it to imaginary circle*, and 
connexion with real cones. It seems all very good, but a 

have no relation to the quaternions. 

* Since you say I must give dates, I must postpone puhli. 

for at least a week beyond next Monday ; and if anyone happens 

VOL. III. 2 D 



4O2 Correspondence between Sir W. Rowan Hamilton and 

to speak to you about my book (London I know has other things 
to think and talk of), you may say, if you choose, that some slight 
causes of delay will prevent it from appearing quite so soon as was 
announced, but that it will almost certainly be out before the end 
of this month.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

* 7, CAM DEN-STREET, CAMDEN TOWN, 
' August 16, 1852. 

* Yours received. I am just come in from Herne Bay, seven 
miles from Canterbury, where I have been reading novels for three 
days, and where I have left my family. I have no doubt your 
ttco solutions were the only ones possible. I was told by the 
inventor, H. Perigal, that there were only two. 

'Never mind the waiting a day or two, or advertisements. 
Dates are of as much importance to an historian as to an Arab. 
The Arab, however, has to dry his ; the historian's are as dr.y as 
possible from the outset.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, August 20, 1852. 

* Without any peculiar call of conscience to do so, I have been 
reading sermons all this morning, an old theological essay of my 
own included, which was printed in the Irish Ecclesiastical Journal 
for (I think) May, 1842. You are not very likely ever to liapjn-n 
to meet that essay, or short Paper, on the Ascension of which I 
could send you a written copy, only that it might look contro- 
versial; but the only controversial hit which it contained was, will 
you believe it, directed against an English clergyman, who after- 
wards came over here, and became my rector and friend if a lay- 
man may speak of his rector and who is now a bishop in England. 
I complained, and (I think) in conversation afterwards convinced 
him, that he had not been orthodox enouy/i. If you ever road my 
Paper you will see that I am quite as Trinitarian as A t liana- 
only I have a different mode of expressing myself. If you 1 
any wish to know my sentiments, you must not assign too much 
importance to my having, in a letter of some mouths ago, atti ilutl 
(I hj>eak from memory) what many persons of my own Chun h 



Professor Augustus De Mot 403 



might consider too little importance to " hypostatical " quo- 
I go with the Catholic Church (that of England and Ireland 
included) on those points ; but do not regard the scholastic aspect 
as the vital one. The influence on the heart and affections, rather 
than on the intellect, seems to me to be the important thing : and 
I am not sure that I shall not enclose a copy of that old Essay of 
my own, with perhaps a sonnet which dared to versify the Te 
Drum, as examples of the in (inner in which Trinitarian doctrines 
may be accepted without injury to charity. Whatever you may, 
in that case, think of my writings on such subjects, I feel quite 
star that you must be pleased with a sermon, of which I think of 
sending you soon a printed copy, by the Rev. George Salmon 
the Mathematician, whose works you know preached at a recent 
Ordination held by the Archbishop of Dublin, and yet in which (I 
think) you could scarcely find a single sentence to object to. (Quito 
otherwise with my Essay you would object to every sentence in 

it.) 

From A. DE MORGAN to SIR W. E. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEX TOWN, 
1 Auyust 23, 1852. 

* I have just finished sorting our friend Baily's correspondence. 
What a man ! He kept and put into the rest, in alpha!' 
order, acceptances of invitations to dinner, apology from the can-Ik' 
man for the defects of the last batch, the printed request of the 
Assurance Office to know if the individual underwritten (who 
happened to be A. De M.) was of good health and sober habits, 
et multa similiora. I have, however, settled what shall be sent to 
Airy to be preserved at Greenwich, and what shall be destroyed. 
One or two letters from you on astronomical subjects are among 
the former. Your handwriting is diminishing as you grow ol<l-r, 
and so I see is mine. We ought all to write a round hand 

now and then, or at least to keep a very broatl-edged pen, an 
I do, and use it some time*, writing large enough to show 
the loops 11 u hi oft I'd. 

* This job being finished, I can find time to turn to you 
your orthodoxy. 

4 1. You say that you are as Trinitarian as Athanasius, and 
r wards [that you go with the Catholic Church. If so, you are 

2D2 



404 Correspondence between Sir W. Rowan Hamilton and 

as Trinitarian as the Athanasian Creed, which is a great deal more 
Trinitarian than Athanasius. 

' 2. Going with the Catholic Church, you do not regard the 
scholastic aspect as the vital one. The Athanasian Creed does, and 
it is the creed of the C. C. 

' 3. In your versification of the Te Deum you Trinitarianize it. 
As it stands no Arian would object to it, provided he might take 
it literally, without any implications from other sources. I have 
often heard it asserted that the Lord's Prayer is Trinitarian. 

* 4. In your sunshine in the Cathedral I am totally puzzled. 
Is it Herschel, Forbes, and Hamilton, who are " in essence one ; 
in name and office three " ? If so, you would be called, if you mean 
the comparison to apply to the Trinity, tritheistic, by all the writers 
on the controversy. If you mean that the Church is in essence, &c., 
I do not understand. I am curious to know what you have read 
on the history of this controversy. 

4 Your " sermon," to which you prophesy the strongest opposi- 
tion on my part, is the only thing against which I have nothing 
to say. I can dispute, on historical evidence, that Athanasius was 
an Athanasian. I can settle by its own creed what the Church of 
England professes. I can dispute the right of anyone to Trini- 
tarianize old formulae made, as I believe, before the Athanasian 
doctrine of the Trinity was in existence ; and I can cry " heresy " 
against any one who understands the Trinity in unity as a cor- 
relative idea to that of the one notion man existing in three different 
men. But your "conjectural" account of the Ascension I am 
wholly without any means of either disputing or confirming. 

* Finally, you are, what very many Churchmen are Roman 
Catholics say, most a high Arian, but you do not know it say a 
high high-Arian ; or, at least, your language bears that construc- 
tion to a person who is so hackneyed in the terms of the controv* 

as myself. I do not think there is much real Athanasian ism in 
the Church of England. But primary and secondary meaning- 
words are so mingled, that two persons who communicate on tlu 
subject need a dictionary of definitions not a chapter. 

1 Here is an adroit play upon a word in its two meanings : 
*B. Robinson, a Baptist minister, well known in the 
century, was in bad odour in his sect, for various heterodox i 
among others it was supposed that he did not admit any corj> 



Professor All oust us DC Morgan. 405 

or personal existence to the Devil. A lady of his sect onoe said 
to him : " Mr. Eobinson, is it really true that you do not believe 
in the Devil ? " " Believe in the Devil, madam," said Robinson, 
" certainly not ; I believe in God don't you ? " 

* I believe I returned you my wife's best thanks for the litho- 
graph of the monody on Sir John Moore. It is a very decisive 
piece of evidence. 

' As you have copied for my amusement, I will copy for 
yours, that you may see what low practical jokes I can descend to. 
I found among Baily's papers a valentine, which I caused to be 
sent to him, in the handwriting of a fair Irish cousin of mine, a 
little while after the publication of the Flamsteed papers. I 
supposed he guessed from whom it came.'* 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, August 26, 1852. 

* Mrs. Flamsteed's letter to Mr. Baily is exquisitely humorous ; 
and I will not show it to anyone who (like Lady Hamilton, to 
whom I have not shown it) might be apt to consider it profane. 

* The particular trinity which I had in view in my Ely sonnet 
was the sisterhood of the three Protestant Episcopal Churches of 
England, Scotland, and Ireland, represented, for the moment, by 
their three respective members Herschel, Forbes, and myself. 
Eightly or wrongly, I took those three Churches as = one Cltun-Ji. 
That the three of us should happen to be kneeling together (quite 
close, in the Dean's pew, or seat, or whatever it is to be call <<!), 
struck me as noticeable, especially as the sudden sunshine im- 
pressed whatever was poetical or excitable in myself at the 
moment ; and falling in with other cathedral images and associa- 
tions, led me to compose the sonnet in question, which, after all, I 
had no particular right to call a " trinitarian " one. 

'It was composed (in 1845) without any pencil or pa; 
while I paced the long cathedral walks after Divine s- 
Herschel and Lady Herschel having also remained in the edifice, 
and being engaged in examining some of the architectural points 
thereof. After some time we met, under the lofty " lantern " ; 

[* See De Morgan's Budget of Paradoxes, p. 188. See also, *//</, p. H.J 



406 Correspondence between Sir IT. Rowan Hamilton and 

and I recited to them my as yet unwritten sonnet of the day. 
After a while Herschel drew me, or, if you will have it so, / drew 
///;;/, into a conversation about the quaternions, at Dean Peacock's 
residence, where we were guests together; and some gentleman, 
whose name I forget, said afterwards that he had heard and 
remembered so much as this of the dialogue, that I had ventured 
to express an opinion to the effect, that "a quaternion, or a function 
of a quaternion, might be the law of the universe." [Unpolarised 
plus polarised intensity = a function of time and space.] Next 
morning, as my bedroom adjoins Herschel' s, and thin partitions 
did my madness from his great wit divide, I early heard what 
Burns might have called a " crooning " ; and was not much sur- 
prised, though of course looking out for a/#, when, before we sat 
down to breakfast at the Deanery, Lady Herschel handed me, in 
her husband's name, and her own for he had contrived to bring 
her into it a sonnet of his to me, which, unless the spirit of egotism 
shall seize me with some unexpected strength, I have no notion of 
letting you see. You did return me the thanks of Mrs. De Morgan 
for the lithograph : I was glad that it gave pleasure.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7, CAMDEN-STREET, CAMDEN TOWN, 
'AugustW, 1852. 

' Tour sheet and proof received. Now tell me the history of 
the facsimile of Wolfe's monody. When was it done ? Where is it 
referred to in the printed discussions on the authorship ? There 
has been a good deal of reference to the subject in the " Notes and 
Queries"; but I have not seen the facsimile referred to. Who 
vouches for Wolfe's handwriting ? 

1 With your letter came one from Peacock, who reports hii: 
restored to health, and able to work " several hours " a-day. ! 
edition of Young's works was nearly finished, and was burnt in the 
fire at Clowes's. It is now being reprinted, so a Peacock is a 
Phoenix. 

* I never should have guessed the solution of the three Chin- 

'In the first place they are not three. " The United Chunh 
of England and Ireland" is one Church not in any sense fw<> 
by article 5 of the Act of Union they are " one Protestant 1 1} i 



Professor Augustus DC Kforym. 4- >; 

pal " Church. The Puseyites, in rejecting the word Protestant, 
have actually shown themselves ignorant of the legal name of their 
own Church. However, not to be guilty of the ferocity of quoting 
an Act of Parliament against a sonnet, what would the High 
Church English say if they knew you have brought the non- 
episcopal church of Scotland into communion with them they, 
of whose churchmanship bishops, priests, and deacons, are the 
essence sometimes the whole essence, for they frequently forget 
the laity altogether ? 

* Your commerce of sonnets (Herschel's and yours) is a curious 
revelation of the inner life of the mathematicians. Who would 
have believed it ? Some day I hope you will send me that sonnet, 
in spite of your determination. Herschel is a man of poetical 
elements, as the American said.' 

From SIR W. R. HAMILTON to A. DE MORGAN. 

'OBSERVATORY, August 31, 1852. 

* I suspect that, as I thought months ago, I must give uj> 
tory perhaps leaving it to you at all events to some future 
writer, who may be myself ; I am too near the subject just now. 

* If you write soon, address to me at Belfast (British Associa- 
tion). Though resolved not to discuss theology with you, I still 
intend to send you Salmon's sermon.' 

From the SAME to the SAME. 

'OBSERVATORY, September 1, : 

* I must have been very obscure, although perhaps not more 
than usually so, in that Ely sonnet, and in my explanation of it. 
Professor (James) Forbes, the one who has experimented on heat, 
and observed the motion of the glaciers, is a Scottish /-./ 

not a Presbyterian. I had no notion of including the " kirk " in my 
little trinity of Churches ; I am far too much a churchman for 
What you say about the English and Irish Protestant Churches 1 
one, by the Act of Union, is quite true; but since in common par 
they are treated as two, and in some practical respects ar 
distinct, I thought myself free in a sonnet to treat them as being 
such. 

' When I was at Edinburgh, in 1850, 1 went to Bishop Torrot's 



408 Correspondence between Sir W. Rowan Hamilton and 

church and heard him preach ; I also formed some acquaintance 
uith him.* The Duchess of Kent attended another Protestant 
Episcopalian Church or perhaps it is to be called chapel, although 
large at the same time. I had been in the same beautiful city 
twice before in 1834 and in 1827. In 1834 I attended, out of 
politeness, at a Presbyterian place of worship; in 1827, I am 
ashamed to say that I cannot remember whether I went to church 
at all. At Belfast, next Sunday, I have some expectation of hear- 
ing my own Archbishop (Whately) preach. He does so, now and 
then, in my own parish church of Castleknock, the same with which 
Swift was connected, if I have not wrongly caught up that story. 
Did you ever hear that I was served, about a year and a-half ago, 
with a " Monition " from the Archbishop's Court, enjoining me, as 
one who had been elected a churchwarden by my co-parishoners 
to remove certain emblems, especially a dove and a pelican, which 
were held by his Grace to be superstitious or unscriptural, from a 
large and handsome stained-glass window, which was, during my 
year of office, erected at the east end of the said parish church, at 
the expense of a pious widow ? An inscription, in which she had 
brought God's name and her husband's somewhat too closely 
together, was also objected to; but, what had scarcely occurred 
to me as likely, the Archbishop was willing to leave in the deceased 
husband's name, if we struck out the name of God. He has, I 
believe, a theory, which perhaps you also hold, that it is Popish to 
regard decorations of houses of worship as contributing to the 
honour of God ; and I counted myself fortunate that, under the 
greater scandal of the window, the present of a handsome organ, 
costing perhaps a couple of hundred pounds, and anonymously 
made by a nephew of my wife while I was a churchwarden, 
escaped without censure or remark. It was rich to see the frightened 
air of the apparitor, when he served me with the Monition ! 
had heard, I suppose, of an un ecclesiastical use of windows, but we 
were on the ground floor at the time; and besides, I was (and am 
far too dutiful a son of the Church to be rude to my Archbishop's 
messenger. So I asked him to sit down, chatted about the forms 



* Bishop Terrot has published, and no doubt discovered for himself, :in 
interesting and curious case of Cotes' s Theorem ; but 1 am sorry to say th.it 1 
have since found it to be anticipated by one of the Bernoulli*. 



Professor Augustus De MorgiDi . 409 

of service, and told him that I had expected some such thing, which 
was the truth. In fact, I flatter myself that, though I had made 
some unsuccessful attempts to divest myself of the office of church- 
warden (which the law of the land, I am told, does not suffer a 
churchman to decline), I was of some r< . the- mutter of the 

window, by acting as a sort of mediator between parties. 

'My rector is a personal friend of the widow, a lady now 
resident in England, but whose husband is buried in our* church- 
yard, and had given, somewhat hastily for he did not even 
consult his own wife ! a permission to have the window erected. 
The Archbishop, somehow, became jealous I won't say that he 
may not have conscientiously disapproved of things in which I 
could see no harm he was my superior and teacher in the affair ; 
but many of us fancied, that if he had been early enough (inked he 
might perhaps have made no objection. My co -church warden was a 
still higher churchman than myself, and had been wishing to present 
our parish church with another painted window. The lady's feelings 
were to be soothed, and a chaplain's importance to be conciliated. 
Altogether there was a precious mess ; and trouble enough I had ; 
but I repeat that I think I was of use. The window stands, an 
emblem or two being removed, according to orders, and the whole 
inscription being boarded over. The lady thanked me for my 
pains, and the Archbishop, last winter, at a party of Lord Claren- 
don's, putting his hand on my shoulder, pronounced me an honest 
man ! apropos to something about Lord Eosse. What I consider 
especially fortunate is, that we all escaped the newspapers ; for so 
much correspondence passed, between one lady and several gentle- 
men, including a prelate, chaplain, rector, two churchwardens, 
and even a lawyer not counting the scribe of the " Monition " 
that when it was all over, another nephew of my wife observed 
that a simple and obvious mode of effacing everything objected 
to in the window would have been to paste over it all the letters 
that had passed ! ' 

* I shall probably be buried there. It would be a luxury, which I have littlr 
right to hope for, to be buried in the College vaults. It I have time to make 
up money for the purpose, I may perhaps arrange to be committal to rarth with 
my fathe'r, mother, and sisters, in the graveyard of St. Mary's, in Dubli: 
Castleknock would do very well. I am quite willing to imagine that the sub- 
ject will bear to be deferred. 



4 IO Correspondence between Sir W. Rowan Hamilton and 



From A. DE MORGAN to SIR W. E. HAMILTON. 

' 7 CAMDEN-STREET, CAMDEN TOWN, 
1 September 1, 1852. 

I meant to have reminded you that printers have a villainous 
habit, still growing, of putting books a year forward, when the 
title-page is printed in the last months of any year. Priority 
will be damaged some of these days by this practice. Pray do not 
only make them put 1852 on the title, but state when the printing 
began. I can bear witness that you meant to be out at Christmas 
last, and that I knew it couldn't be. 

' Remember delivery of MSS. to printer is publication, to say 
nothing of delivery of proofs and fair sheets to me. 

* I like your determination not to discuss theology. You send 
paragraph or sonnet, and I send answer Voila la discussion com- 
mencee. As the proverb says, " The fray begins at the second 
blow." You mean that you withdraw the record after the defen- 
dant's answer, and issue a new writ, e. g. Salmon's sermon yet to 
come. Mem. I will read it, even if it be of the n th order. 

' Now your theology brought back into my head the old lines 
which were written on a Bible, at the time when private judgment 
began to develop itself 

' Hie liber est in quo quserit sua dogmata quisque, 
Invenit et pariter dogmata quisque sua. 

And then there came into my head a paraphrase 

* One day at least in every week 

The sects of every kind 
Their doctrines here are sure to seek 
And just as sure to find. 

From all this you will gather that I have received the fair sli 
2 Y I think. 

* I hope your trip to Belfast will do you good. For m\ 
solitude is my relaxation. When my family go away to the sea, 
and all the people leave town, and I sort all the letters, and put 
everything to rights, I feel what I used to hear called an ayr< 



Professor Augustus De MOJ 4 1 1 



From SIR W. B. HAMILTON to A. 

'OB8KK7ATOKY. 1852. 

4 1 wrote a long letter (two sheets of note-paper) to you yester- 
day, yet believe that I did not say a word about tho lithograph of 
the ode on the death of Sir John Moore, by Charles Wolfe. It 
is very easy for me, however, to give you full satisfaction 01 
subject. The lithograph was published in 1841, in the Proc 
of the Eoyal Irish Academy, the original letter of Wolfe having 
been produced before that body on the 26th of April in that year, 
during my Presidency, but while my friend Humphrey Lloyd was 
occupying the chair in my absence. It was Dr. Anster, the Irish 
translator of Faust, who produced the letter or fragment, -which h;ul 
been found by Dr. Luby, now Senior Fellow of T.C.D., among the 
papers of a deceased brother of his own, who was a college friend 
of the writer and receiver of that letter. At this moment I speak 
literally " from book," namely, from Part V. of the Pro* 
the E. I. A., for the year 1840-41, where the lithograph is inserted, 
between pages 88 and 89. But as I am on cordial terms with 
Doctors Lloyd, Anster, and Luby (Luby, author of a Triyonon 
&c.), it would cost me nothing to make a little farther inquiry, if you 
could possibly think it worth while. Dr. Anster remarked to the 
Academy that it was not his object to treat seriously the " insane 
pretensions, now and then put forward in the newspapers for this 
person or the other," but rather to furnish, and place on record, an 
authentic copy of the ode, from the text of which some deviations 
have been made in print : such as the erroneous substitution of 
"suddenly" for "sullenly" firing. Wolfe was before my tinio, 
but I am acquainted with his biographer, Archdeacon 
and a deceased friend of mine, the Rev. Dr. Samuel O'Sullivan, 
who knew a vast deal about Irish life and society in a now past 
generation, often told me that he was with Wolfe, during a } 
the time when the poetical afflatus was upon him, in connexion 
with the subject of the ode. 

* I ought to be in Belfast now, and hope to be so to-morrow. 
My Preface is a horrid bore to me. There is so much t ha 
say, and so little time or even space for saying it, and I so \ 
wish to do justice to other people, while I have so little h 
being able to do it. In short I think that I must just cut the 



412 Correspondence between Sir W. Rowan Hamilton and 

knot, by one desperate sword-stroke, and come forward as a dis- 
coverer, perhaps, but not as an historian. 

' Your joke about the " dates " amused Lady Hamilton.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 



' 7, CAMDEN-STREET, CAMDEN TOWN, 
< September 3, 1852. 



' I understand you at last. Prop. It is legitimate, at least for 
poetry, to consider the trinity as analogous to three churches 
in three different countries. But there is an older proposi- 
tion. You remember the story of the mathematician returning 
Paradise Lost with " What does it prove " ? He might easily have 
been answered. It proves that it is impossible to introduce the 
Trinity into poetry without either Tritheism or Arianism. When 
Milton's recently- found book proved him to have been formally 
Arian, the learned world (at least the reviewing world) said : " We 
always thought so ; look at Paradise Lost." They had no right to 
think so. Paradise Lost is but one of two things, either Arian or 
Tritheistic by poetical necessity. 

* Now for the church window. I would bet five to one (in Os, 
they are my rouleaux) that Whately was right. Emblems are 
what they are understood to mean. So says common sense ; so says 
the law. And the pelican is notoriously and distinctively Roman. 
The organ is not superstitious, because it suggests not more of one 
side than another in fact suggests nothing except what the player 
makes it suggest for the time being. 

* I do not hold with the Archbishop that " it is (distinctively) 
Popish to regard decorations as to the honour of God." It is also 
Greek and English, to different extents. I believe it to be BUJ 
etitious (which Popish is perhaps a synonym for in your senten 
but there are Protestant superstitions which are not Popish) ; but 
if I were able, by holding up a finger, to abolish superstition mid 
mysticism, I would not. I should as soon take the hot manure 
away from a plant before it was fit to draw its nourishment from 
the soil. Religion unmixed with superstition does not assimilate 
with the mental organs, and will not until a far higher stat- 
cultivation is attained. But we must keep what approach we have 
made, and not fall back, and therefore the pelican must be aboli>lu-l. 



Professor Augustus DC Mor^in . 413 

* Not only churchmen, but dissenters, are bound to serve as 
churchwardens, and dissenters not unfrequently do serve in Eng- 
land. And without going to church during service time. 

'My notions about burial would shock a poet. If I had quite 
my own way in the matter, I should say, " Let the machine in 
which I have done duty be carried to those whose business it is to 
mend it while in action, that they may, by examination of it, 
become better qualified to mend other machines. If you want 
some of my remains to perform a ceremony over, take any pair of 
old breeches you find in my drawers. As soon as I have ceased to 
think whether by that which thought leaving the body, or by 
thought undergoing a temporary annihilation the remains of the 
animals which fed me, and the remains of the animals which 
covered me, only differ in this that the former were worked up 
by a much more skilful workman than the latter ; and therefore 
the former are worth preserving for examination, while the latter 
may be buried if you please." Now I will give you time to re- 
cover from the horror you will feel at this bit of religion without 
superstition/ 

From SIR "W. E. HAMILTON to A. DE MORGAN. 

* HOOP HILL HOUSE, LURGAX, 

* September 1, 1S.VJ. 

' In the midst of packing for a second change of position (a 
practical provection what would Donaldson of the Cratylus say ?), 
I just write this line to acknowledge that I received at least two or 
three notes from you at Belfast, where I made a mathematical com- 
munication which the local newspapers characterised as "exclusively 
scientific," and, therefore, not to be reported. Perhaps from my 
next place of rest, near the beautiful bay of Carlingford this 
place is near Lough Neagh I may be able to write to you.' 

From the SAME to the SAME. 

' CASTLE HOUSE, CABLINGFORD, 

< Septcmlt'i- 17, 1852. 

' The morning before I left the Observatory for Belfast, some 
of my children asked me at breakfast, as they had often done 
before, how the Quaternions were getting on. To which my 
answer was, Hang the Quaternions ! (I assure you I had so much 



414 Correspondence between Sir W. Rowan Hamilton and 

propriety left as not to say D them ! ) But now, after some 

pretty complete diversion of thought, for a while, from a subject 
which had occupied me, perhaps too much, I am beginning to 
feel an interest in it again ; and to-day, before and after a sail of 
three or four hours on the lovely bay (of Carlingford) which, when 
I raise my head, I see, I quite enjoyed reading some of my own 
articles on definite integrals in quaternions, while walking in an 
old friend's garden here the friend whose guest I am. Several 
other acquaintances, in Carlingford and Rostrevor, claim a little of 
my time on this occasion of my visiting this neighbourhood, and I 
do not now expect, though anxious to get home, to start as early as 
Monday. Therefore, if you are inclined to write me a line 
though I have not yet answered your letters to Belfast please to 
direct it to Carlingford (the place of oyster celebrity, as it this 
moment comes into my head to remember). It is not unlikely 
that I may write to you again to-morrow, partly about the winding- 
up of my " publication " (in the common sense of that word ; 
yours, I am sure, is the correct one), and partly about other things/ 

From A. DE MORGAN to SIR W. R. HAMILTON. 

7, CAMDEN-STREET, CAM DEN TOWN, 
1 September 20, 1852. 

* Just been reading extracts from an Irish journal on the Duke 
of Wellington, and his dislike (alleged) of his own country. 
Query emerges given X - land, what constitutes an X -man. The 
Duke was of a descent English within the memory of man ; settled 
in Ireland : his mother Irish : at what period begins complete 
Hibernicism ? 

* I thank you (as I suppose) for the Belfast newspaper. The 
meeting seems to have thriven. 

4 1 see that Lord Rosse's name is among those quoted for tin* 
Chancellorship of Oxford. Surely he would not trouble himself 
with such a post. As soon as the Dons get a man who would not 
awe them, they would pull him to pieces with their feuds. 

* This town is so empty that it supplies no news. 

'I forget if I asked you before who I. Walton was. lie wai 
the Dublin opponent of Berkeley's Analyst, and published thi 
tracts in the very year of the Analyst. I have a suspicion he \\ 
the son of old Isaac, but I cannot verify either way.' 






Professor A u git si us De Morga)i . 415 

From SIR W. E. HAMILTON to A. DE MORGAN. 

1 OBSEBVATOBT, WEAR DIBLIM, 
' September 25, 1852. 

* This note is to telegraph my safe arrival here, after a 1 
tour of three weeks in the north of Ireland, which has done my 
health and spirits so much good, that Lady Hamilton says I 
41 ought to go very often to Carlingford." While there I wrote, 
and at Kostrevor (on the opposite side of the bay) corrected and 
cent off to London last Tuesday, a little Paper for the Phil. Mag., 
of which I received, and have examined, a proof this morning. 

' P. S. I had written your name in a copy of Salmon's 
Tinon before starting, and shall look for it presently in the 
De Morgan basket, but cannot stay to do so at this moment.' 

From A. DE MORGAN to SIR W. E. HAMILTON. 

1 7, CAMDEN-STREKT, r AMI.KN TOWK, 
1 September 27, 1852. 

* You and your Eostrevor note probably travelled to Dublin 
together, and the note and your Dublin note also travelled 
together, and arrived together this morning. So your want of 
strategics has cost you a penny. As you are come home in wild 
health and spirits you will probably not care, but put it down to 
travelling expenses. 

1 Your basket reminds me (did I tell you ?) that I have had 
fifty little portfolios made for 3c/. a-piece. I dare say you could 
get them in Dublin for 2d. Nothing but two sheets of thin 
pasteboard, 4to size, with three bits of book- covering cloth or paper- 
imitation (the young people could make them beautifully) pasted 
on, so as to open out wedge wise, but with parallel planes (this is 
important). Accordingly, I have arranged all my 1 
alphabetical order. But H is a heavy letter even without you and 
your namesake ; so you have a portfolio to yourself. You h;i\ 
no idea what a provocative to order and decency is the having a 
number of little receptacles ready. 

* I have just been getting up (for Phil. Mag.} the early hi- 
of infinitesimals in England. Few of us know that Leibni:. 
perfectly well known in England before the dispute, and that 



416 Correspondence between Sir W. Rowan Hamilton and 

Newton's first provocative to an imperfect publication was ds and 
infinitely small quantities paraded under his own eyes by an 
English writer (Craig), who lent him his MSS. to read. 

' What of Quaternions ? I too may ask ; but I suppose you are 
really on eve of publication. 

* Don't forget Isaac Walton of T. C. D., the Irish assailant of 
Berkeley; he is much unknown. Misspelt Wilson in Thomson's 
History of Royal Society. Mr. Salmon is on the spot, I think, to 
look at records. I will read his sermon as I said before.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

1 OBSEBVATOBY, September 29, 1852. 

' You remind me of sending you Salmon's sermon : the 
pamphlet itself I had not forgotten ; on the contrary, I took it out 
of your basket yesterday, and read it for my own satisfaction, 
pencilling, from memory, in the middle of page twenty, the word 
0v<no7. Just now I have asked my boys, and their tutor, but with- 
out success, to find in the Greek Testament the passage referred to, 
which I remembered to run thus : 17 yvwmc $ud(ot, 17 Se ay airy OIKO- 
cofiti. I have just found it for myself, in 1 Cor. viii. 1. Perhaps, 
in this instance, my memory is more of the heart than of the head ; 
for I remember the passage being quoted to me in conversation, 
considerably above twenty years ago, with Salmon's (probably 
received) interpretation, as potential, not indicative, by one whose 
blood yet cries from Irish earth against his murderers : the Tiev. 
Mr. Whitty, who was stoned to death while returning from a work 
of mercy. I do not mean of proselytism, with which he had not 
much to do. I have not read Salmon's sermon on Temporal 
Blessings, but believe so far in National Sins and Judgment- 
to think (whatever my Archbishop may), that Irish miseries are nut 
unconnected, judicially, with Irish murders. It will be a genuine 
satisfaction to me, if though it is too good a thing almost to li 

for Fathers B - and C shall be transported, for tlio 

murderous assault on the Queen's troops, which they encouraged 
at Six-Mile-Bridge. At present I have no sympathy to spin 
the criminal assailants in that business, who met with so drs 
a punishment. 

* Will you believe, after this, that I do not even /m-// all the 



Professor Augustus De Morgan . 417 

popish priests to be hanged, or even banished? On the coir 
on giving, about a year ago, a Douay Bible (not Testament), AS 
by-the-way, cost me something not inconsiderable much more 
than a good copy of the authorized version- to a Roman Cir 
female in this house, I told her to consult her priest as to reading it : 
and was informed that he gave her perfect permission to do BO. 
Indeed, on a hint conveyed through her, the said priest, 1 
Dungan somewhat of the older and better school of Romish 
ecclesiastics waited on me soon afterwards, and accepted a 1 
for the temporal use of his poor parishioners ; that stipulation being 
made by me, on the ground that I was a Protestant myself. 
" Well, Sir," said he, " perhaps you cannot help it ! " His charity 
led him, as you see, to cloak me with "invincible ignorance." 
But perhaps I have told you all this before. If I do so a third 
time, let me know. 

I At the time of the potato famine I refused to assist the same 
priest in some expressly popish enterprise ; but offered him money 
for temporals, which he, but quite kindly and politely, declined. 
At that time I gave away money more indiscriminately than I now 
choose to do, and he pointed to a crowd of people who were wait- 
ing at my door. Still I fancy that the priests are now much 
poorer than they were a few years ago. 

I 1 have heaps of things to say, about biquaternions, &c., &c.* 

From A. DE MORGAN to SIR W. R. HAMILTON. 

' 7, CAMDEN- STREET, CAMDEN Towx, 
'September 29, 1852. 

' Please to send me name and address of your printer. My 
publisher will, perhaps, have to ask him for an estimate for a work 
not of mine. It is, in fact, Boole who is meditating typography 
on his mathematical logic, which is a very original tiling, and, for 
power of thought, worthy to be printed by the printer of the 
Quaternions. As he has Hibernicised himself, it is fit he should 
have an Irish printer, and I rather suspect that the Dublin man 
prints more cheaply than the London ones. 

* You are not, perhaps, aware of the check which there is i; 
employing distant printers. It is the difficulty of being quite sure 
that they do not take off more than the stipulated iiuml 

VOL. III. 2 E 



4 1 8 Correspondence between Sir W. Rowan Hamilton and 

copies, and sell the overplus for their own benefit. The thing has 
been done. Of course respectable men can be trusted not to do 
this : the difficulty is to know the respectability. But there is no 
fear in the case of mathematical logic. Could you contrive a 
check ? I see none but the publisher furnishing the paper, and 
having it made for the work with a peculiar water- mark/ 

From the SAME to the SAME. 

1 7, CAMDEN-STREET, CAMDEN TOWN, 
'October 1, 1852. 

6 1 sent yesterday a missive for the name and address of your 
printer. Yours received in the evening. Salmon's argument that 
God intended religion as a means of developing the inquisitive 
faculties is, I think, very sound. But it follows that differences of 
opinion were intended, for a unanimous uninfallibility would be 
just as drowsy a dormitory as an infallible Church. Therefore 
you and I are to differ in opinion, quod erat inveniendum, or, 
whether or no, quod fuit inventum. Therefore also Salmon and 
I are to differ, which we do on one point. He thinks he may 
deduce something from the Epistle to Timothy containing no 
esoteric doctrines. I think not. An epistle was too public a thing 
in those days; and Paul must needs know that Timothy would 
have to show, and probably permit copy of, his letter. 

' I agree with you in wishing that the fathers of the Six-Mile- 
Bridge controversy may be transported if the matter can be proved 
against them. 

* All Catholics, noiv, introduce the salvo of " invincible igno- 
rance." This must be a modern doctrine, surely, since it is clear 
that they did believe that all the heathens even those whose 
ignorance was never assailed were fated to everlasting perdition. 
'On looking at a tract of Walton (about whom I have a stand- 
ing inquiry at the Irish in general), I find in it an answer \ 
tract of Berkeley, of which I have never heard a thinl trn< 
the Analyst controversy. I never heard of more than tiro of 
It is entitled Reasons for not rcplt/incj to Mr. Walton * full an> 
Surely T. C. Library must have everything of Berkeley. 

4 As for Ireland, trust to the Exodus the best thing that 



Professor A u gust us De Morga n. 419 

happened. I did in one sense forestall the spirit of Archbishop 
Whately's pun as recorded in the newspapers. I avowed my 
-delight that the Irish had got into Exodus, because there had been 
a great deal too much Genesis. 

' When you gave the pound to the priest for temporal purport, 
how do you know that you did not set free another poun 
spiritual ones? The only offer I ever made to subscribe to a 
B. C. fund was when a friend of mine (a new convert) informed 
me that a subscription was making to build Cardinal Wiseman a 
palace. I offered to be at the expense of painting the street door, 
provided I might choose the colour and the number. This offer 
was not accepted.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, October 3, 1852. 

' I have not at this moment your last note beside me, but r- 
that I have let a couple of days pass without answering it, although 
so many others have been left much longer, and are even still, 
unanswered. You inquired the name and address of my printers, 
or printer. He remonstrated with me, about a year ago, for not 
using the latter form of grammar : being probably what is called 
a " warm " man, and very probably far better off than I am in the 
world ; but at all events being jealous lest I should fancy him to 
have " partners," in a concern which by honest industry he has 
gradually made his own, namely, the press of this Dublin Univer- 
sity. When I wish to be polite, I address him, by post, as 
follows : 

' " M. H. Gill, Esq., University Press Office, Dublin." 

* You can judge already, from what you have seen of my sheets, 
respecting the fitness of his establishment to conduct scieir 
printing; and I know that he professes to be ready to print in 
most languages, and, with a little time for preparation, to promise 
to print in any. In short, he is ambitious of excellence in his 
own business; has printed for Salmon, Jellett, and others in 

[hematics, and (I think) for Hincks, in those investigations 

pecting the strangely deciphered characters of the ancient East. 

With Inniy if I were again correcting the impression of a passage in 

Apollonius, referred to in a note to the Preface of my Essay on 

2E 2 



420 Correspondence between Sir W. Rowan Hamilton and 

Pure Time, I should not be obliged to argue, or insist, that it made- 
some difference whether a Beta or a Theta were used in a Greek 
quotation. 

* It may be a foolish shyness, but although I am much better 
acquainted with Boole than with you, as far as talk goes, for he 
took tea with me at my temporary rooms in Christ Church, Ox- 
ford, in 1847, along with Adams and (I think) Win. Thomson, 
and has since met me (or I him) at a grand dinner of doctors in 
Dublin, whereas I think that I have only met you once in my life ; 
yet still, the habit of correspondence makes so great a difference in 
the feeling of freedom, that I should, on the whole, prefer your 
merely sending to him the enclosed slip, to your letting him see 
this note, however little private in any important sense it can be. 

' P. S. The quotation was from Euclid, after all, and the word 
c/3aAAoju*vrj, which the then printers of the E. I. A. persisted for 
some time in making ficSaAAojulvij ; and on my going personally 
to remonstrate, gravely assured me that it was of no consequence ! * 

From the SAME to the SAME. 

* OBSERVATORY, October 4, 1852. 

' Another note has reached me this morning, and I find that I 
can answer your query about Walton, or at least about the tract of 
Berkeley which you mention, without consulting the Library of 
T. C. D., which, from distance, is nearly useless to me. Berkeley's 
" Reasons for not replying to Mr. Walton's Full Answer, in a 
Letter to P. T. P.," are printed at page 49 (to page 62) of the 
third and last volume of a work of which the following is a copy 
of the title-page : " The Works of George Berkeley, D.D., late 
Bishop of Cloyne in Ireland. To which is added an Account of i 
his Life ; and several of his Letters to Thomas Prior, Esq., Dean 
Gervois, Mr. Pope, &c. In three volumes. London : printed ly 
,J. F. Dove, St. John's Square; for Eichard Priestly, 14^, Bigh 
liolborn, 1820." 

4 Tlio Tract is bitterly ironical, as you may judge from the two 
last sentences of its first section, or division. " But those, Sir, : 
not the reasons I shall assign for not replying to Mr. Walton's full 
answer. The true reason is, that he seems at bottom a fa<rti 
man, who under the colour of an opponent writes on my Mi- 
llie question, and really believes no more than I do of {Sir 



Professor Augustus De Morgan. 42 1 

Newton's doctrine about Suctions (sic), which he exposes, contradict*, 
and confutes, with great skill and humour, under the mask of a 
grave vindication." [The word " fluxions " seems to be elsewhere 
.spelled in the usual way, throughout the Tract.] My copy of the 
work is not mine (like Berkeley, you know, I am an Irishman) ; it 
belongs to the Earl of Dunraven ; but if you cannot otherwise get 
& sight of it, he would probably let me lend it to you. Of Walton 
himself I know nothing, except what may be collected from the 
two tracts of Berkeley : the " Appendix," and the " Reasons." 
About my own Appendix, more anon. Yours again.' 

From A. DE MORGAN to SIR W. R. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEN Towir, 
' October 5, 1852. 

* I see you are on bibliographical P's and Q/s with me, but you 
nave forgotten to state whether your edition of Berkeley, which is 
quite new to me, is 4to, 8vo, folio, 12mo, or 18mo or what. I 
will not borrow it, for I shall find it at the British Museum. I 
had Berkeley in two vols., 4to, London, 1784; but the idea of 
space which they communicated was so large that, as they had no 
mathematical works, I was obliged to sell them. I am very much 
obliged to you for the reference to this edition. I used to deal 
with Priestley, in Holborn, but I never saw this edition exposed 
in his shop. 

' Your phrase, " my copy is not mine," is not a bull. It is 
perfectly good English to use the same word in two different 
senses in one sentence, particularly when there is usage. An 
English farmer said : " I did not get so much for these calves as 
I expected ; and I never thought I should." To expect is said of 
the price demanded : " How much do you expect ask for these 
calves " ? Incongruity of language is no bull, for it expresses 
meaning. But incongruity of ideas (as in the case of the Irish- 
man who was pulling up the rope, and finding it did not finish, 
oried out that somebody had cut off the other end of it) is the 
genuine bull. 

* The late Dr. Thomson of Glasgow (the chemist), always spelt 
fluxions." 

' I shall forward the printer's name to Boole. I shall be 



422 Correspondence between Sir W. Rowan Hamilton and 

glad to see his work out, for he has, I think, got hold of the true- 
connexion of algebra and logic.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

1 OBSERVATORY, October 8, 1852. 

* The mere look of this sheet may serve to distinguish it from 
my usual notes, sufficiently to help your memory in case of your 
being disposed to read, at some indefinitely future period, what it 
may contain respecting some recent investigations of mine, without 
troubling yourself about them at present.' 

[This is the commencement of a letter of 12 folio pages. The 
conclusion was posted on the 28th of October. It related to Gauche 
Polygons, a subject treated of by Hamilton in communications to 
the K. I. A. in 1849, 1850, 1853. See also Lectures on Quaternions,. 
p. 309.] 

From A. DE MORGAN to SIR W. E. HAMILTON. 

* 7, CAMDEN-STREET, CAMDEN TOWN, 
1 October 23, 1852. 

* I am trying a fine pen \vith which to write in books. I think anyone 
would suppose 1 was a small thin man, to look at the results. I have received 
sheet i, possibly to be returned. This index speaks of pages up to 719. I have 
not received beyond 704. Your unfinished letter is not a bore ; when the 
commencement of the session is terminated, I intend to approfound it. 

1 1 not only found the edition of Berkeley you speak of, but another, luti-r, 
edited by G. N. Wright, London, 1843, two vols., 8vo. It is singular that two- 
editions of Berkeley should have been so recent, and hardly anybody have heard 
of them. The more so, as Wright says some liberties were taken with Berkeley 1 ! 
text in the quarto edition. 

'Having given the nibbler a fair trial, I now resume my 
ordinary pen. I shall send you, in a few days, a paper on tin 1 
early history of infinitely small quantities in England. It is but 
a little specimen of the suppressions which national controvt 
gives rise to. From the moment when Newton declared against 
infinitesimals (1704), which till then he had exclusively used in 
fluxions, the English world agreed to suppose that they never hud 



Professor Augustus DC Morgan. 423 

been used here, and to forget the works in which they had been 
used. All these works are now absent from the Royal Society 
library, except Newton's P rind [tin. 

6 A student of mine asked me to-day to give him a reason for a 
fact which I did not know was a fact, and do not yet. He says, 
that if a figure be anyhow divided, and the compartment* 
differently coloured, so that figures with any portion of common 
boundary line are differently coloured four colours may be 
wanted, but not more. Query cannot a necessity for five or more 
be invented? As far as I see at this moment, if four ui . 
compartments have each boundary line in common with one of the 
others, three of them inclose the fourth, and prevent any fifth from 
connexion with it. If this be true, four colours will colour any 
possible map, without any necessity for colour meeting colour 
except at a point. 

'Now, it does seem that drawing three compartments with 
common boundary, two and two, you cannot make a fourth take 
boundary from all, except by inclosing one. But it is tricky 
work, and I am not sure of all convolutions. "What do you say ? 
And has it, if true, been noticed ? My pupil says he guessed it 
in colouring a map of England. The more I think of it, the more 
evident it seems. If you retort with some very simple case which 
makes me out a stupid animal, I think I must do as the Sphvnx 
did. If this rule be true, the following proposition of logic 
follows : 

'If A, B, C, D, be four names, of which any two might be 
confounded by breaking down some wall of definition, then some 
one of the names must be a species of some name which includes 
nothing external to the other three.' 

From SIR W. B. HAMILTON to A. DE MORGAN. 

< OBSERVATORY, October 26, 1852. 

' I am not likely to attempt your " quaternion of colours " very 
soon. 

' The difficulty about that horrid " Preface " of mine is the 
historical one. If you were in the humour, and found time, to send 
me any sketch of your own, of the history of " Double Algebra," 
or even of the history of your own extensions and improvements 



424 Correspondence between Sir W. Rowan Hamilton and 

thereof, I should be only too happy to insert such sketch in an 
Appendix. But that is probably too good a thing to hope for. 
Some time or other I trust that you mil publish some historical 
account of our friend -v/^I, going back, by all means, to the 
Hindoos about whom I know less now than when I was a boy. 
Meanwhile I shall give you one little bit of modern confession or 
record on the subject, which naturally interests myself. 

* In page xiv of the Preface (Yorrede) to Grassmann's Ausdeh- 
nungslehre, a very original work, published (it appears) at Leipzig, 
in 1844 Verlag von Otto "Wigand which work, if any, the 
Germans, if they think me worth noticing, will perhaps set up in 
rivalship with mine, but which I did not see till long after my own 
views were formed and published. In that page xiv Grassmann 
says, under date " Stettin der 28, Juni 1844," . . . " Eben so 
nun zeigte sich auch umgekehrt, wie man vermittelst der so 
gefundenen Bedeutung des Imaginaren auch die Gesetze der 
Analyse innerhalb der Ebene ableiten kann, hingegen ist es nicht 
mehr moglich, vermittelst des Imaginaren auch die Gesetze/wrflfefi 
Raum abzuleiten. Auch stellen sich iiberhaupt der Betrachtung 
der Winkcl im Rawne Schwierigkeiten* entgegen, zu deren allseitiger 
Losung mir noch nicht hinreichende Musse geworden ist." 

' The underpointings are my own. But I think that I must 
cite this passage, because Grassmann has a non-commutative 
principle of multiplication of his own, although the foregoing 
passage (dated June, 1844) seems to prove that he had not then (nor, 
so far as I have learned, since] the notion, much less the laws of 
the quaternion, which I communicated to the E. I. A. in October 
and November, 1843, and of which an outline was printed in the 
Philosophical Magazine for July, 1844.' 



* I did not consider myself to have so much as begun the geometrical theory of 
the Quaternions for an algebraical (or metaphysical) theory of ete had been Lnj, p 
previously in my head till I had (as I thought, and still think) conquered those 
difficulties for angles in space, which Grassmann avows to have been inextricably 
perplexing him when " publishing " several months later, in 1844. 



Professor A tig list us De Morgan . 425 



From A. DE MORGAN to SIR W. H. UAMILT 

7, CAMDEN-STIIEET, CAMDEW Town, 
' October 29, 1852. 

* To write anything worthy of your book at the beginning of a 
Session is not at all on the cards for me. Moreover, I should think 
you might well content yourself with a well-dated account of your 
own doings, and of the analogous ones, so far as they touch triplets 
or quaternions. The doublets you might assume. You would thus 
enable others to secure your own rights, and Grass-mann (quere, is 
his Christian name Nebuchadnezzar ?) would fall at once into his 
proper place. The thing most requisite is a faithful account of 
what you had seen up to publication I mean not a full account of 
all the books, but general notice. 

'Ex. (jr. when I wrote my Logic I gave an account, in general 
terms, of all I knew of any attempt to deal with the syllogism, 
otherwise than Aristotelically. From this it appears that I knew 
of one work of Ploucquet. Your Scottish synonyme, in his recent 
work, finding himself unable to refer to his charge of pillaging 
himself, finds out, or thinks he finds out, some of my ideas in other 
works of Ploucquet, which he mentions, and thereupon charges me 
with pillaging Ploucquet, whose writings (the man uses the plural) 
he says, I admit I had read. Now I am safe, and the charge is un- 
worthy of any notice, for all who believe me commonly honest (and 
argument from me will be of no use with others) will see that I 
knew of but one work of Ploucquet, and that one not one of those 
which Sir W. H. cites as containing the matter in question. 

* By the way, when you send me German, clarify it into 
English. For I belong to the school which reads German with a 
dictionary, and guesses the syntax ; or, if it be mathematics, makes 
the equation translate the paragraph immediately preceding it. If 
you cite the passage, translate it.' 

From SIR W. E. HAMILTON to A. DE MORGAN. 

'OBSERVATORY, October 38, 1852. 

f At last I have fairly dashed into my Preface- writing, 
decided to continue, although as briefly as I can, a sketch which 1 
began before I went to Belfast, of the progress of my own thoughts 



426 Correspondence between Sir W. Rowan Hamilton and 

on the whole subject, and especially to give a very abridged account 
of my old view of Pure Time, taking great pains, however, to 
avoid your old (and not ill-natured) charge of " dogmatism," by 
not pretending that mine is the only view which can be taken 
of the matter. As it really was the parent of the quaternions, so 
I have, over and over again, returned to the opinion, resisted 
through unwillingness to seem to insist too much on my own old 
speculations, that some general knowledge of the point of view, 
from which I looked at algebra long ago, would be a really useful 
preparation for a student of my later speculations on geometry. 
He would thereby see how natural it was for me to use, in the 
latter science, some notations which at first seem strange ; and other 
advantages, I think, would follow. 

' You may keep the unre vised half -sheet of Contents ; but I 
wish you to return the enclosed slips of an Appendix (C), which 
will be found somewhat stiff (I suspect) by readers for some time 
to come. It records, in a condensed form, the leading steps of a 
long analysis ; and the condensation has cost me more trouble, I 
think, than the original investigation did. On the same principle, 
the composition of the Table of Contents was perhaps the most 
laborious part of my work/ 

From A. DE MORGAN to SIR W. E. HAMILTON. 

1 7, CAMDEN-STREET, CAMDEN Tcnvx, 
1 November 8, 1852. 

' I got your note, and I think I got from you a newspaper 
to-day, about the Historical Society, for which thanks. I am glad 
to hear that you mean to give your own history at least, and that 
is all which can be claimed. I am up to my ears in series ainl 
mean values. I believe I have put the subject on the same footing 
as any question of limits. You take the Philosophical Maynziif . 1 
think. If you do, it is not worth while to bother you with sepaniU' 
copies of Papers in it, unless you bind tracts methodically.' 

From SIR W. R. HAMILTON to A. DE MORGAN. 

' OBSERVATORY, November 16, IS 

* My manuscript, for slips five and six, which is already in the 
compositor's hands, begins with the word " triplets " ; but a note 



Professor Augustus De Morgan. 427 

has been appended to that word, viz. : " These remarks on triplets 
are now for the first time published." I think that the) 
interest you, as showing how far I had gone in February, It* 

From the SAME to the SAME. 

' OBSERVATORY, November 18, 18.' 

4 You need not trouble yourself to return the enclosed slip. I 
shall ask the printers to send me another copy. 

* When I get slip six, and forward you a copy of it, you will 
find some equations transcribed from a manuscript-book of Febru- 
ary, 1835, respecting the precise date of which it is impossible for 
me to have any doubt whatever. Even the place where I wrote 
the investigation is recorded in that old manuscript ; it was Kilboy, 
the seat of Lord Dunalley, near Nenagh, who is not distantly con- 
nected with my wife's family, and who, although he has never h.i 1 
occasion or opportunity to show more than civility to her or me, 
exhibited in former years his sense of relationship by very substan- 
tial proofs, such as purchasing commissions in the army for more 
than one brother of hers since dead. If it were needful, or 
becoming, I could attest in a court of justice the genuineness and 
date of the manuscript-book from which I have quoted, as having 
had nothing added to it since the month already mentioned. But 
I hope that few if any persons will think me such a/oo/, as to set 
up, at this time of day, any claim grounded on that manuscript, 
especially as against you. That would be too ridiculous ; and 
though I can bear ridicule, I do not court it. As soon as your 
Paper on Triple Algebra (of which you sent me a proof-sheet) was 
published, I felt that the time for gracefully claiming the posses- 
sion of triplets was lost to me. Accordingly you may have noticed 
that I have been since extremely guarded and cautious in my 
printed allusions to the subject. See, for instance, the last page of 
my quarto Essay '(First Series of Researches on Quaternions), pub- 
lished in 1847. But it seems to me that if I now put on record 
some old conceptions and results of my own, to which I have more 
than once in print alluded, ev